BACKSTEPPING, in its infinite-dimensional version, has

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST Backstepping-Forwarding Control and Observation for Hyperbolic PDEs With Fredholm Integrals Federico Bribiesca-Argomedo, Member, IEEE and Miroslav Krstic, Fellow, IEEE Abstract An integral transform is introduced which allows the construction of boundary controllers and observers for a class of first-order hyperbolic PIDEs with Fredholm integrals These systems do not have a strict-feedback structure and thus the standard backstepping approach cannot be applied Sufficient conditions for the eistence of the backstepping-forwarding transform are given in terms of spectral properties of some integral operators and, more conservatively but easily verifiable, in terms of the norms of the coefficients in the euations An eplicit transform is given for particular coefficient structures In the case of strictfeedback systems, the procedure detailed in this paper reduces to the well-known backstepping design The results are illustrated with numerical simulations Inde Terms Boundary control, boundary observation, Hyperbolic PDEs, integral transform I INTRODUCTION BACKSTEPPING, in its infinite-dimensional version, has proven to be a very effective tool for constructing boundary controllers and observers for large classes of PDEs, see for instance 1 1, with numerous applications such as: control of turbulent flows 11, boundary control of the Korteweg-de Vries Euation 12, output tracking on heat echangers 13, delay compensation for finite-dimensional systems 14, and electrochemical battery models 15 Nevertheless, the use of a Volterra transform restricts the class of systems to which it can be applied (they must have a strict-feedback structure) Recently, some results have appeared for specific classes of systems with non strict-feedback components In particular, results are available for finite-dimensional systems with either distributed delays or some PDE in the actuation or sensing path that gives it a non strict-feedback structure, 16, 17 and certain other PDE structures, see 18 In this article, we present an integral transform of the state of a PIDE that allows us to build a stabilizing boundary control for a class of first-order hyperbolic PIDEs with Fredholm integrals (non-strict feedback terms) that arise, for instance, when considering coupled PDE-ODE or PDE-PDE systems with boundary actuation in only one of the euations Manuscript received February 6, 214; revised August 2, 214; accepted January 21, 215 Date of publication February 2, 215; date of current version July 24, 215 Recommended by Associate Editor D Dochain F Bribiesca-Argomedo was with the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA USA and is now with Université de Lyon-Laboratoire Ampère, (CNRS UMR55)-INSA de Lyon, Villeurbanne Cede, France ( federicobribiesca@insa-lyonfr) M Krstic is with the Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA USA ( mkrstic@ucsdedu) Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 1119/TAC Specifically, we consider systems of the form ū t (, t) =ū (, t) d()ū(, t) f()ū(,t) ḡ(, y)ū(y, t)dy h(, y)ū(y, t)dy, (, t) (, 1) (,T (1) ū(1,t)=ū(t), t (,T (2) with initial condition ū(, ) =ū () L 2 (, 1; R) Where d, f, ḡ and h are real-valued continuous functions in their respective domains Using the change of variables u(, t) =e d(ξ)dξū(, t) (3) proposed in 2, we can focus without loss of generality on the stabilization of the euation (without reaction term) u t (, t) =u (, t)f()u(,t) g(, y)u(y, t)dy h(, y)u(y, t)dy, (, t) (, 1) (,T (4) u(1,t)=u(t), t (,T (5) with initial condition u(, ) = u () L 2 (, 1; R) With f, g and h real-valued continuous functions in their respective domains, and boundary control U(t) For the observer design, we consider u(,t) to be the only available measure The coefficients f, g, and h can be epressed in terms of those appearing in (1) as f() = d(ξ)dξ e f() (6) g(, y) = d(ξ)dξḡ(, e y y) (7) h(, y) = e y d(ξ)dξ h(, y) (8) This class of systems is related to that presented in 2, however, the possible presence of non-strict-feedback terms (whenever h is not zero) means that it cannot, in general, be stabilized using a backstepping approach The two integral terms appearing in (1) can be thought of as a Fredholm integral with a piecewise-continuous kernel, possibly having a discontinuity at y = The dependence of the kernel on makes the problem more challenging but, at the same time, more relevant (as illustrated by the eamples presented) IEEE Personal use is permitted, but republication/redistribution reuires IEEE permission See for more information

2 2146 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 The control problem tackled in this article is then to find a gain kernel γ C(, 1; R) such that, under the control law U(t) = γ(y)u(y, t)dy (9) the origin of system (4), (5) is finite-time stable in the topology of the L 2 norm The observation problem in turn, is formulated as a stabilization problem for the error system (the difference between the estimated and real states) and an adeuate output error injection gain γ obs,1 C(, 1; R) must be found The results presented in the first part of Section II (up to Section II-E) are an etended version of those presented in 19 including complete proofs and a reworked simulation eample They concern the general form of the euation and provide different conditions for a stabilizing boundary controller to eist In particular, concrete conditions on the magnitude of the coefficients in (4) will be given which are sufficient for a solution to eist and for it to be given as the limit of a given seuence The approach presented in the second part of Section II (starting with Section II-F), on the other hand, restricts the class of systems under consideration by adding supplementary assumptions on the shape of the coefficients in (4) that allow the computation of an eplicit controller gain for the system Finally, Section III tackles the observer design problem II BACKSTEPPING-FORWARDING CONTROL DESIGN A Preliminary Definitions In order to build a stabilizing controller for system (4), (5) we proceed by finding a bounded transform w(, t) =u(, t) with bounded inverse u(, t) =w(, t) and the associated control law U(t) = p(, y)u(y, t)dy k(, y)w(y, t)dy (, y)u(y, t)dy (1) l(, y)w(y, t)dy (11) p(1,y)u(y, t)dy (12) such that system (4) is mapped into the (finite-time stable) target system w t (, t) =w (, t), (, t) (, 1) (,T (13) w(1,t)=, t (,T (14) A more precise formulation of the transform will be given after the necessary spaces are defined It will be shown (the proof can be found in Appendi A) that the kernels of the direct transform need to satisfy the following condition: p (, y)p y (, y) =g(, y)(, 1)p(1,y) y y h(s, y)p(, s)ds g(s, y)p(, s)ds g(s, y)(, s)ds,, y, 1 st y, y (15) (, y) y (, y) =h(, y)(, 1)p(1,y) with boundary condition p(, ) = f() y y h(s, y)(, s)ds g(s, y)(, s)ds h(s, y)p(, s)ds,, y, 1 st y (16) p(, y)f(y)dy (, y)f(y)dy,, 1 (17) In general, a second boundary condition is reuired for these euations to be well defined In this section we choose to impose (, 1) = which will simplify the contraction arguments reuired in the proofs by eliminating the nonlinear terms in (15) and (16) A somewhat different procedure is presented in Section II-F since the particular structure of the considered kernels reduce the system of PDEs to a first-order (nonlinear) ODE in the spatial variable, for which the condition on p(, ) is epressed as k 1 () = The resulting ODE is already well defined (under some assumptions) so the boundary condition corresponding to (, 1) is not reuired Furthermore, an eplicit solution can be obtained for this ODE The boundedness of the direct transform (as an operator mapping between adeuate normed vector spaces) implies that any bounded initial condition of the original system corresponds to a bounded initial condition of the target system The boundedness (again, as a map between adeuate normed vector spaces) of the inverse transform implies that, as the norm of the target system goes to zero, so does the norm of the state of the original system Therefore, the eistence of both a bounded direct and inverse transforms imply the stability of the original system in some function space The natural choice of the function spaces in which to define the direct and inverse transforms (and thus the stability results)

3 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2147 will depend on the regularity of the obtained kernels In this article we focus only on obtaining continuous kernels The procedure reuired to obtain higher regularity is analogous and more cumbersome Definition 1: Let us define two (closed, bounded) subsets of R 2 as follows: euipped with the norm T l = {(, y), 1, 1,y } (18) T u = {(, y), 1, 1, y} (19) z =ma{ z1, z 2 }, z =(z 1,z 2 ) R 2 where denotes the absolute value of an element of R (NB whenever necessary, we consider R to be euipped with the topology induced by the absolute value metric, or the euclidean norm in R) We should note that (T l, ) and (T u, ) are compact in the topology induced by their norms Hereafter, unless otherwise eplicitly stated, we assume T l and T u to be euipped with these norms Furthermore, the chosen norm is euivalent to the usual Euclidean norm Definition 2: We now define the spaces X l = C(Tl ; R) and X u = C(Tu ; R) euipped with the norm Xl (respectively Xu ) defined as s Xl =sup s(z), s X l (2) z T l s Xu =sup s(z), s X u (21) z T u Note that (X l, Xl ) and (X u, Xu ) are Banach spaces These are the spaces in which we will define the kernels in our integral operators Definition 3: Given functions φ X l, ψ X u we define the operator Π φ,ψ : L 2 (, 1; R) L 2 (, 1; R) as Π φ,ψ ξ() = φ(, s)ξ(s)ds ψ(, s)ξ(s)ds (22) for all ξ L 2 (, 1; R), and all, 1 Based on this definition, we can write the transforms in (1) and (11) as and w(, t) =(I L 2 Π p, )u(,t) () (23) u(, t) =(I L 2 Π k,l )w(,t) () (24) for all (, t), 1,T, where I L 2 is the identity operator on L 2 (, 1; R) Assumption 1: The coefficients in (4) satisfy: f C(, 1; R), g X l and h X u Definition 4: Define now the space euipped with the norm X = X l X u (25) ϕ X =ma { ϕ1 Xl, ϕ 2 Xu }, ϕ =(ϕ1,ϕ 2 ) X (26) As defined, (X, X ) is a Banach space We now introduce an integral operator T related to the PDEs the kernels in (1) must satisfy in order to map the dynamics of (4) to those of (13) Definition 5: Define the integral operator T : X X (for A 1,1 : X l X l, A 1,2 : X u X l, A 2,1 : X l X u, A 2,2 : X u X u, F 1 X l F 2 X u ), for all p X l, X u as where T p A 1,1 p(, y) = = p A F A1,1 A = 1,2 y y σ A 2,1 A 2,2 f(s)p( y, s)ds y y p F1 F 2 h(s, σ)p(σ y, s)ds dσ g(s σ, σ) (27) p(σ y, σ s)ds dσ (28) 1y A 1,2 (, y) = f( y s)( y, y s)ds y 1σy g(σ y s, σ) (σ y, σ y s)ds dσ (29) A 2,1 p(, y) = 1y σ h(s, σ y)p(σ, s)ds dσ (3) A 2,2 (, y) = 1y y h(s σ, σ y) (σ, σ s)ds dσ 1y 1σy g(s σ y, σ y) (σ, σ y s)ds dσ (31) F 1 (, y) = f( y) F 2 (, y) = 1y y g(σ y, σ)dσ (32) h(σ, σ y)dσ (33) in their respective domains Net, we introduce an integral operator R related to the conditions reuired for (11) to be a left-inverse of (1) This operator is obtained by substituting (1) into (11) Definition 6: Given functions φ X l, ψ X u, we define an operator R φ,ψ : X X as R φ,ψ k = S φ,ψ k φ l l ψ S φ,ψ 1,1 S φ,ψ 1,2 k φ = S φ,ψ 2,1 S φ,ψ (34) l ψ 2,2

4 2148 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 where S φ,ψ 1,1 k(, y) = S φ,ψ 1,2 l(, y) = S φ,ψ 2,1 k(, y) = S φ,ψ 2,2 l(, y) = y y ψ(s, y)k(, s)ds y φ(s, y)k(, s)ds (35) φ(s, y)l(, s)ds (36) ψ(s, y)k(, s)ds (37) ψ(s, y)l(, s)ds y φ(s, y)l(, s)ds (38) in their respective domains Finally, we define an operator related to the PDE conditions that the kernels of (11) must verify in order to map the dynamics of (13) into those of (4) Definition 7: Define also the integral operator T : X X (for Ā 1,1 : X l X l, Ā 1,2 : X u X l, Ā 2,1 : X l X u, Ā 2,2 : X u X u, F 1 X l F2 X u ), for all k X l, l X u as where T k l Ā 1,1 k(, y) = y = k Ā l F Ā1,1 Ā = 1,2 Ā 1,2 l(, y) = Ā 2,1 k(, y) = y Ā 2,1 Ā 2,2 y 1σy k F1 l F 2 (39) g(σ y, s σ)k(s σ, σ)ds dσ h(σ y, s σ y) k(s σ y, σ)ds dσ (4) y y σ f(σ y)l(,σ)dσ 1y 1yσ g(σ y, s)l(s, σ)ds dσ (41) h(σ, s σ y) k(s σ y, σ y)ds dσ (42) 1y Ā 2,2 l(, y) = f(σ )l(,σ y)dσ 1y σ 1y y g(σ, s)l(s, σ y)ds dσ h(σ, s σ ) l(s σ, σ y)ds dσ (43) F 1 (, y) = y f( y) F 2 (, y) = 1y in their respective domains g(σ y, σ)dσ (44) h(σ, σ y)dσ (45) B Direct Transform Proposition 1: If the operator T, as defined in (27), has a uniue fied point in X (ie, there eists a uniue ζ X st Tζ = ζ), then transform (1) with kernels p = ζ (46) maps system (4) (9), with γ(y) = p(1,y), y, 1 into (13), (14) The proof of this result is given in Appendi A An euivalent condition to that of Proposition 1 is that 1 belongs to the resolvent set of the operator A, as defined in (27) For the conditions reuired for a value to belong to the spectrum (or the resolvent) of a bounded operator on a Banach space the reader is directed to 2, Lemma 1213 Using Banach s contraction mapping principle, see for eample 21, Theorem 31, we can establish sufficient conditions for the previous results to hold Corollary 2: If the operator T, as defined in (27), is a contraction then transform (1) with kernels p = lim T n ϑ (47) n for any ϑ X, maps system (4) (9), with γ(y) = p(1,y), y, 1 into (13), (14) In particular, if T is a contraction, it implies that the spectral radius of A is less than 1 (and therefore 1 does not belong to the spectrum of A) Even though this condition is conservative, it allows for a constructive result to be given (the kernels can be found using Picard iterations) Particularly noteworthy is the fact that this corollary depends on the choice of norm used in the definition of the Banach space X A similar result can be obtained whenever there eists a positive integer n for which T n is a contraction However, since the computations become etremely cumbersome after more than a couple iterations (ecept for very particular cases) we only give the proofs for the case where T is a contraction mapping Using the supremum norm, associated to our space X, we can give a sufficient condition in terms of the magitude of the coefficients in (4) for the direct transform to eist It should be noted that this bound is conservative since few conditions are imposed on the coefficients For some particular cases it can be easily relaed (for instance, if f() =this bound is doubled) Lemma 3: If the coefficients in (4) verify c =ma{sup s,1 f(s), g Xl, h Xu }<(1/2), then transform (1) with kernels p = ζ = lim T n ϑ (48) n

5 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2149 for any ϑ X, maps system (4), (5), with γ(y) = p(1,y), y, 1 (49) into (13), (14) Furthermore ζ X F X 1 2c (5) Proof: If we can show that there eists C, 1) such that Tϕ T ϕ X C ϕ ϕ X, ϕ, ϕ X (51) then the operator T is a contraction We start by noting that Tϕ T ϕ X = Aϕ A ϕ X, ϕ, ϕ X (52) and Aϕ A ϕ X = A(ϕ ϕ) X (53) Let us denote K = ϕ ϕ X, and c defined as in the theorem statement, then after some computations we obtain the norm estimate { } A(ϕ ϕ) X ma ck sup (1y),cK sup (1y) (54) y,1 y,1 which in turn implies A(ϕ ϕ) X 2cK (55) If 2c < 1, T defines a contraction mapping The application of Banach s contraction mapping principle 21, Theorem 31 completes the first part of the proof The norm estimate comes from rewriting ζ = A n F (56) n= and noting that it implies, using (55) ζ X F X n= (2c) n (57) This epression and the condition c<(1/2) complete the proof C Inverse Transform In this section we focus on the computation of the inverse transform (assuming the direct transform has already been obtained) The first results use the definition of the operator R p, to give conditions for the left-inverse of the direct transform to eist Similar conditions can be found for its right-inverse and it can be shown, using the associativity of linear operators from a space to itself, that if the left- and right-inverse eist they are eual Where necessary, this condition is given in terms of the spectrum of the operator Π p, Proposition 4: Given kernels p X l and X u,ifthe operator R p,, as defined in (34) has a uniue fied point ϕ X, then transform (11) with kernels k = ϕ (58) l is the left-inverse of transform (1) The proof of this Proposition follows by applying first the direct and then the inverse transform to an arbitrary function in L 2 (, 1; R) and reuiring the result to be the original function A condition euivalent to that in the Lemma is that 1 belongs to the resolvent set of the operator S p,, as defined in (34) After applying Banach s contraction mapping principle, the following corollary is obtained: Corollary 5: Given kernels p X l and X u, if the operator R p, as defined in (34) is a contraction, then transform (11) with kernels k = lim l n (Rp, ) n ϕ (59) for any ϕ X, is the left-inverse of tranform (1) Using the norm estimate obtained in Lemma 3 we obtain the following sufficient condition for the eistence of an inverse transform (left- and right-inverse): Lemma 6: If the coefficients in (4) verify ma{sup s,1 f(s), g Xl, h Xu } < (1/4), then for kernels p X l and X u as defined in Lemma 3, transform (11) with kernels k = lim l n (Rp, ) n ϕ (6) for any ϕ X, is the inverse of tranform (1) Furthermore, the operator Π p, defined in (22) has a spectral radius less than 1 Proof: Applying Lemma 3, the condition in this result implies that the direct transform eists and that the operator T has a uniue fied point (since the norm of the coefficients is less than 1/2) The stronger 1/4 bound on the coefficients reuired here, together with the norm estimate at the end of Lemma 3, implies that R p, is a contraction and that Π p, has an operator norm less than one, which implies that (I L 2 Π p, ) is boundedly invertible (and thus its left- and right-inverse is the same) Finally, using Corollary 5 we obtain that (11) is the leftinverse of (1) and must therefore be its inverse This completes the proof Repeating the procedure in Proposition 1 but mapping from the target system to the original one, we obtain an operator T analogous to the previously considered operator T In practice, the Picard iterations for this operator converge more easily than those of R p, and therefore the following conditions may be easier to test: Lemma 7: If the coefficients in (4) verify ma{sup s,1 f(s), g Xl, h Xu } < (1/2), then for kernels p X l and X u as defined in Lemma 3, if the uniue fied point of T is also the fied point of R p, and 1 belongs to the resolvent set of Π p,, then transform (11) with kernels k l = lim n ( T ) n ϕ (61) for any ϕ X, is the inverse of tranform (1) Proof: Following a procedure analogous to that used in the proof of Lemma 3, the condition on the coefficients implies that T is a contraction and therefore has a uniue fied point Furthermore, by a similar procedure to the one used in the proof of Proposition 1, we obtain that the transform (11), with the kernels given by the fied point of T maps system (13), (14) into (4) (9) The condition that T is also the fied point of R p, guarantees that (11) is the left-inverse of (1) and, since 1 belongs to the resolvent set of Π p, it is also the right-inverse thus completing the proof

6 215 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 This formulation ensures that T is a contraction and therefore does not reuire the spectral radius of S p, to be less than 1 The resulting conditions on the coefficients are weaker than those needed for R p, to be a contraction and the result can, therefore, be more easily applied We must stress that reuiring one of these operators to be a contraction is not necessary for the backstepping-forwarding techniue to work but guarantees that Picard iterations can be used to find the necessary fied points of the operators D Closed-Loop L 2 Stability The previous sections gave conditions for the direct and inverse transforms to eist In this section we present the first main result in this paper Proposition 8: If 1 belongs to the resolvent set of the operators A defined in (27) and Π p, defined in (22), with kernels p =(IX A) 1 F (62) then the origin of system (4) (9), with γ(y) = p(1,y), y, 1 is finite-time stable in the topology of the L 2 (, 1; R) norm Proof: The first condition in the Theorem guarantees, by Proposition 1, that transform (1) is bounded and maps system (4) (9), with γ(y) = p(1,y), y, 1 into (13), (14) The second condition guarantees that the inverse transform eists and is bounded 2, Lemma 1213 Finally, the finite-time convergence to zero of the state of the target sytem (13), (14) completes the proof A conservative (but easy to verify) sufficient condition for the above result to hold is: Theorem 9: If the coefficients in (4) verify that ma{sup s,1 f(s), g Xl, h Xu } < (1/4) then the origin of system (4) (9) is finite-time stable in the topology of the L 2 (, 1; R) norm, with γ(y) = p(1,y), y, 1 where p = ζ = lim T n ϑ (63) n for any ϑ X Proof: The conditions in this result imply, by Lemma 3, that the direct transform eists and maps (4) (9), with γ(y) = p(1,y), y, 1 into (13), (14) Lemma 6 completes the proof As was the case in the inverse transform, a more practical condition to verify may be: Proposition 1: If the following conditions are verified: (i) the operator T defined in (27) is a contraction in some norm euivalent to X and therefore has a uniue fied point ζ X, (ii) the operator T defined in (39) is a contraction in some norm euivalent to X and therefore has a uniue fied point ϑ X, and (iii) setting p = ζ (64) 1 belongs to the resolvent set of Π p, and ϑ is the fied point of R p, then the origin of system (4) (9), with γ(y) = p(1,y), y, 1 is finite-time stable in the topology of the L 2 (, 1; R) norm Proof: Conditions (i) and (ii) are set in order to find the fied points of T and T using Picard iterations (they give directly a constructive solution method for the resulting kernel integral euations) As a direct conseuence, since 1 belongs to the resolvent set of Π p,, the transform (1) is invertible and, ϑ being the fied point of R p,, by Proposition 4, its inverse is given by (11) with k = ϑ (65) l E Application to a PDE-ODE Interconnected System Consider the following first-order PDE coupled with a second order ODE: u t (, t) =u (, t)au(,) bv(, t) (66) =v (, t) cv(, t)du (, t) (67) with a, b> and boundary conditions u(1,t)=u(t) (68) v (,t)= (69) v(1,t)= (7) This system closely resembles the Korteweg-de Vries-like euation presented in 2 The only two differences (other than notation) are the addition of a (destabilizing) term au(,t) and the use of only one boundary to control the full interconnected system (instead of using one boundary of each subsystem) Solving (67) with boundary conditions (69), (7) and plugging the resulting epression into (66), we obtain a representation of the form (4) with f() =a bd sinh ( c(1 )) (71) c cosh( c) g(, y) = bd cosh( c)cosh( c(1 y)) cosh( c) bd cosh ( c( y) ) (72) h(, y) = bd cosh( c)cosh( c(1 y)) cosh( (73) c) We now present simulation results for a =125, b =1, c =1, d =1 For these coefficients, a solution can still be found for both systems of integral euations (even though they are larger than the sufficient condition presented in Theorem 9) and therefore the direct and inverse transforms eist and are bounded Fig 1(a) and (b) show the obtained direct (respectively inverse) transform kernels for this system Fig 1(c) shows the obtained control gain Fig 2 shows the evolution of the state in open-loop (unstable) and closed-loop (finite-time stable) F Eplicit Boundary Controller With Shape Restrictions in the Coefficients In this subsection, we impose additional conditions on the structure of the coefficients in (4) and the transform kernels

7 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2151 Fig 2 Simulated evolution of the open-loop and closed-loop behavior of the u(, t) state of the interconnected PDE-ODE system (a) Open-loop evolution of the PDE state u(, t) (b) Closed-loop evolution of the PDE state u(, t) for f 1, λ R, with boundary condition u(1,t)=u 1 (t) (75) for all t (,T This restricted form, along with the assumptions that follow (reuired only in this subsection) will allow us to find an eplicit epression for the controller and its associated transform Assumption 2: h 1 () is such that Fig 1 Direct and inverse transform kernels obtained numerically for the interconnected PDE-ODE system and resulting control gain (a) Direct transform kernels p(, y) and (, y) (b) Inverse transform kernels k(, y) and l(, y) (c) Control gain p(1,y) in order to obtain an eplicit solution to the nonlinear kernel (15), (16) with boundary condition (17) With this structure (restricting the degrees of freedom for the kernels), the boundary condition on (, 1) used in the previous section is no longer reuired to obtain a well-posed system under certain assumptions In this subsection, we will restrict the general class of systems (4), (5) to the more particular form 1 h 1 (s) s e α(ys) f 1 e λy dy ds (76) where α = λ f 1 eλy h 1 (y)dy Theorem 11: If Assumption 2 is verified, then the origin of the system (74), (75), with control where k 1 is given by: U(t) =f 1 e λ k 1 (y)u(y, t)dy (77) u t (, t) =u (, t)f 1 e λ h 1 (y)u(y, t)dy, (, t) (, 1) (,T (74) k 1 () = eα(s) h 1 (s)ds 1 h 1(s) s eα(ys) f 1 e λy dy ds is (finite-time) stable in the topology of the L 2 norm (78)

8 2152 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 k1 den = e α ( 3α 3 (2 3e e 3 )α 2 3(1e e 3 )α 2e 3 6e 2 ) 6e 2 α (81) and α =(1/3)(2 3e e 3 ) III BACKSTEPPING-FORWARDING OBSERVER DESIGN A Observer Structure For any practical implementation of the controllers constructed in the previous section, the construction of an observer is reuired We now turn to the observer design problem for a first-order hyperbolic system with the same structure as (4) and measured output u(,t) We propose the following observer structure: û t (, t)=û (, t)f()û(,t)γ obs,1 ()û(,t)u(,t) g(, y)û(y, t)dy h(, y)û(y, t)dy, (, t) (, 1) (,T (82) û(1,t)=u(t), t (,T (83) with initial condition û(, ) =û () L 2 (, 1; R) Here, γ obs,1 () is a gain to be determined The resulting error system is given by ũ t (, t) =ũ (, t)f()ũ(,t)γ obs,1 ()ũ(,t) g(, y)ũ(y, t)dy h(, y)ũ(y, t)dy, (, t) (, 1) (,T (84) ũ(1,t)=, t (,T (85) where ũ(, t) =û(, t) u(, t) We can define γ obs () = f()γ obs,1 () and focus only on the backstepping-forwarding stabilization of the error system Fig 3 Simulated evolution of the open-loop and closed-loop behavior of the u(, t) state of the PIDE (a) Open-loop evolution of the PDE state u(, t) (b) Closed-loop evolution of the PIDE state u(, t) (c) Control gain f 1 e λ k 1 () The proof of this result is given in Appendi B G Numerical Eample of Eplicit Controller Fig 3 shows the open-loop and closed-loop behavior under simulation of a system of the form (74), (75) with f 1 =2, λ =2 and h 1 () =cosh() The corresponding eplicit controller is k 1 () = knum 1 () k1 den (79) with k num 1 ()= 3(α 2)e α( αe α α cosh()sinh() ) (8) ũ t (, t) =ũ (, t)γ obs ()ũ(,t) h(, y)ũ(y, t)dy, g(, y)ũ(y, t)dy (, t) (, 1) (,T (86) ũ(1,t)=, t (,T (87) B Preliminary Definitions In order to build an observer for system (86), (87) we proceed by finding a bounded transform ũ(, t) = w(, t) k obs (, y) w(y, t)dy l obs (, y) w(y, t)dy (88)

9 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2153 with bounded inverse w(, t) =ũ(, t) and the associated gain p obs (, y)ũ(y, t)dy obs (, y)ũ(y, t)dy (89) γ obs () =k obs (, ) (9) such that the error system (86) is mapped into the (finite-time stable) target system w t (, t) = w (, t), (, t) (, 1) (,T (91) w(1,t)=, t (,T (92) We remark that, for the observer design, we proceed by first finding the transform mapping from w to u and then its inverse (mapping from u to w) Assumption 1 is maintained throughout this section Analogously to the control case, the kernels of the inverse transform for the observer need to satisfy a set of PDEs k obs, (, y)k obs,y (, y)=g(, y)k obs (, )l obs (,y) y y g(, s)l obs (s, y)ds g(, s)k obs (s, y)ds h(, s)k obs (s, y)ds,, y, 1 st y, 1 (93) l obs, (, y)l obs,y (, y)=h(, y)k obs (, )l obs (,y) with boundary condition y y g(, s)l obs (s, y)ds h(, s)l obs (s, y)ds h(, s)k obs (s, y)ds,, y, 1 st y (94) k obs (1,y)=, y, 1 (95) In this section, a second boundary condition l obs (,y)=is chosen to cancel the nonlinearity in the kernel PDEs and simplify the contraction arguments reuired to solve the euations First, we introduce an integral operator R related to the conditions reuired for (89) to be a left-inverse of (88) Definition 8: Given functions φ X l, ψ X u, define an integral operator R φ,ψ : X X as R φ,ψ pobs obs = φ ψ S φ,ψ pobs obs (96) with the operator S φ,ψ defined as in (34) We now introduce an integral operator T obs related to the PDEs the kernels in (88) must satisfy in order to map the dynamics of (86), (87) to those of (91), (92) Definition 9: Let us now define the integral operator T obs : X X (for A obs 1,1 : X l X l, A obs 1,2 : X u X l, A obs X u, A obs 2,2 : X u X u, F obs l obs X u as where T obs kobs l obs = Aobs A obs 1,1 k obs (, y) = 2,1 : X l 1 X l F 2 X u ), for all k obs X l, kobs l obs A obs 1,1 A obs 1,2 = A obs 2,1 A obs 2,2 1 y F obs kobs l obs g(σ, s σ y) F obs 1 F2 obs k obs (s σ y, σ y)ds dσ 1 1σ h(σ, s σ ) (97) k obs (s σ, σ y)ds dσ (98) A obs 1,2 l obs (, y) = 1 σy g(σ, s) l obs (s, σ y)ds dσ (99) A obs 2,1 k obs (, y) = 1σy h(σ, s σ y) A obs 2,2 l obs (, y) = k obs (s σ y, σ y)ds dσ (1) σ y g(σ, s)l obs (s, σ y)ds dσ h(σ, s σ) l obs (s σ, σ y)ds dσ (11) 1 F1 obs (, y) = g(σ, σ y)dσ (12) F obs 2 (, y) = in their respective domains h(σ, σ y)dσ (13)

10 2154 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 Finally, we introduce an integral operator T obs related to the PDEs the kernels in (89) must satisfy in order to map the dynamics of (91), (92) to those of (86), (87) Definition 1: Define the integral operator T obs : X X (for Āobs 1,1 : X l X l, Ā obs 1,2 : X u X l, Ā obs 2,1 : X l X u, Ā obs 2,2 : obs obs X u X u, F 1 X l F 2 X u ), for all p X l, X u as pobs = pobs T obs Ā obs obs F obs obs where = Ā obs 1,1 p obs (, y) = Āobs 1,1 Ā obs 2,1 1 σy 1 y Ā obs 1,2 Ā obs 2,2 pobs obs F obs 1 F 2 obs (14) h(s, σ y)p obs (σ, s)ds dσ g(s σ y, σ y) p obs (σ, σ y s)ds dσ (15) Ā obs 1,2 obs (, y) = 1 1σ g(s σ, σ y) Ā obs 2,1 p obs (, y) = obs (σ, σ s)ds dσ (16) σ Ā obs 2,2 obs (, y) = h(s, σ y)p obs (σ, s)ds dσ y h(s σ, σ y) (17) obs (σ, σ s)ds dσ 1σy g(sσ y, σ y) obs (σ, σ y s)ds dσ (18) 1 F 1 obs (, y) = g(σ, σ y)dσ (19) F obs 2 (, y) = in their respective domains h(σ, σ y)dσ (11) C Direct Transform For the observer, the direct transform (88) maps the target system to the original error system (contrary to the control case) For the eistence of the direct transform we have the following results (analogus to those for the control) Proposition 12: If the operator T obs, as defined in (97), has a uniue fied point in X (ie there eists a uniue ζ X st T obs ζ = ζ), then transform (88) with kernels kobs = ζ (111) l obs maps system (91), (92) into (86), (87), with γ obs () = k obs (, ),, 1 (112) The proof of this result is analogous to that in Appendi A and is omitted for brevity An euivalent condition to that in Proposition 12 is that 1 belongs to the resolvent set of the operator A obs, as defined in (97) We give a sufficient condition on the coefficients for the results to hold: Lemma 13: If the coefficients in (86) verify c obs = ma{ g Xl, h Xu } < 1, then transform (88) with kernels kobs = ζ l obs = lim n T n obsϑ (113) for any ϑ X, maps system (91), (92) into (86), (87), with and γ obs () = k obs (, ),, 1 (114) ζ X F obs X 1 c obs (115) The proof is analogous to that of Lemma 3 and is therefore omitted It should be noted that the conditions in this section are somewhat less stringent than those used for the control design This is due to the fact that, for the observer design, u(, ) is measured and, therefore, the coefficient f() can be compensated perfectly D Inverse Transform In this section we focus on the computation of the inverse transform (assuming the direct transform has already been obtained) The first results use the definition of the operator R k obs,l obs in (96) to give conditions for the left-inverse of the direct transform to eist Similar conditions can be found for its right-inverse and it can be shown that if the left- and rightinverse eist they are eual Where necessary, this condition is given in terms of the spectrum of the operator Π k obs,l obs Proposition 14: Given kernels k obs X l and l obs X u,if the operator R k obs,l obs, as defined in (96) has a uniue fied point ϕ X, then transform (89) with kernels pobs obs = ϕ (116) is the left-inverse of transform (88) A condition euivalent to that in the previous Proposition is that 1 belongs to the resolvent set of the operator S k obs,l obs, as defined in (34) Using the norm estimate obtained in Lemma 13 we obtain the following sufficient condition for the eistence of an inverse transform (left- and right-inverse):

11 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2155 Lemma 15: If the coefficients in (86) verify ma{ g Xl, h Xu } < (1/2), then for kernels k obs X l and l obs X u as defined in Lemma 13, transform (89) with kernels 1 belongs to the resolvent set of Π k obs,l obs fied point of R k obs,l obs then the origin of system (86), (87), with and ϑ is the pobs obs = lim n ( R k obs,l obs ) n ϕ (117) for any ϕ X, is the inverse of tranform (88) Furthermore, the operator Π k obs,l obs defined in (22) has a spectral radius less than 1 The proof is analogous to that in Lemma 6 E Closed-Loop L 2 Stability The previous sections gave conditions for the direct and inverse transforms to eist In this section we present the main observation result Proposition 16: If 1 belongs to the resolvent set of the operators A obs defined in (97) and Π k obs,l obs defined in (22), with kernels kobs =(IX A obs) 1 F obs (118) l obs then the origin of system (86), (87), with γ obs () = k obs (, ),, 1 (119) is finite-time stable in the topology of the L 2 (, 1; R) norm The proof is analogous to that of Proposition 8 and is therefore omitted A conservative (but easy to verify) sufficient condition for the above result to hold is: Theorem 17: If the coefficients in (86) verify that ma{ g Xl, h Xu } < (1/2) then the origin of system (86), (87) is finite-time stable in the topology of the L 2 (, 1; R) norm, with γ obs () = k obs (, ),, 1 (12) where kobs = ζ l obs = lim T n n obsϑ (121) for any ϑ X The proof is analogous to that of Theorem 9 and is therefore omitted Again, a more practical version of the results is: Proposition 18: If the following conditions are verified: the operator T obs defined in (97) is a contraction in some norm euivalent to X and therefore has a uniue fied point ζ X, the operator T obs defined in (14) is a contraction in some norm euivalent to X and therefore has a uniue fied point ϑ X, and setting kobs = ζ (122) l obs γ obs () = k obs (, ),, 1 (123) is finite-time stable in the topology of the L 2 (, 1; R) norm The proof is analogous to that of Proposition 1 and is therefore omitted F Stability of Observer and Controller In this section, we discuss the stability of the observer and controller interconnection This means we consider systems (4), (5) and (82), (83) with U(t) = p(1,y)û(y, t)dy and initial conditions u (), û () L 2 (, 1; R) We assume that kernels p, k obs X l and, l obs X u are given satisfying (15) (17) and (93) (95) We further assume kernels k, p obs X l and l, obs X u are given such that (I L 2 Π k,l ) is the inverse of (I L 2 Π p, ) and (I L 2 Π p obs, obs ) is the inverse of (I L 2 Π k obs,l obs ) Using the definition of ũ(, t) =û(, t)u(, t), stability of (u, û) is euivalent to stability of (u, ũ) We therefore focus on (4), (5) and (86), (87) with U(t) = p(1,y)u(y, t)dy p(1,y)ũ(y, t)dy and initial conditions u (), ũ () =û () u () L 2 (, 1; R) Applying the backstepping-forwarding transformations, we change variables to w(, t) =(I L 2 Π p, )u(,t)() and w(, t) =(I L 2 Π p obs, obs )ũ(,t)() The transformed system dynamics are given by w t (, t) =w (, t) w(1,t)= (, 1)p(1,y) (I L 2 Π k obs,l obs ) w(,t) (y)dy (124) p(1,y)(i L 2 Π k obs,l obs ) w(,t)(y)dy (125) w t (, t) = w (, t) (126) w(1,t)= (127) with initial conditions w () =(I L 2 Π p, )u () and w () =(I L 2 Π p obs, obs )ũ () L 2 (, 1; R) These euations can be solved as w(, t) w (t) t (σ, 1)p(1,y) (I L 2 Π k obs,l obs ) w(,tσ)(y)dy dσ, for t 1 = p(1,y)(i L 2 Πk obs,l obs ) w(,t1)(y)dy (σ, 1)p(1,y) (I L 2 Π k obs,l obs ) w(,tσ)(y)dy dσ, for t>1 { (128) w ( t), for t 1 w(, t) = (129), for t>1 for all t,, 1 Using Hölder s ineuality, the boundedness of the kernels (in the X l or X u norm, respectively), and the boundedness

12 2156 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 Fig 4 Resulting control and observer gains (a) Control gain γ(y) =p(1,y) (b) Observer gain γ obs () =k(, ) of (I L 2 Π k obs,l obs ) as an operator in L 2 (, 1; R) it can be shown that there eists a constant C(p,, k obs,l obs ) > (ie, depending only on p,, k obs, and l obs ) such that the norm estimates w(,t) L 2 w L 2 (13) w(,t) L 2 w L 2 C(p,, k obs,l obs ) w L 2 (131) hold for all t Furthermore w(,t) L 2 =, t 1 (132) w(,t) L 2 =, t 2 (133) These norm estimates guarantee the stability of the interconnected system and the finite-time convergence in 2 seconds of the transformed state (w, w) Furthermore, together with the boundedness of (I L 2 Π k,l ) and (I L 2 Π k obs,l obs ) it implies that there eist positive constants C 1, C 2, C 3 depending only on p,, k, l, p obs, obs, k obs, and l obs such that for all t, and ũ(,t) L 2 C 1 ũ L 2 (134) u(,t) L 2 C 2 u L 2 C 3 ũ L 2 (135) ũ(,t) L 2 =, t 1 (136) u(,t) L 2 =, t 2 (137) Fig 5 Simulated evolution of the closed-loop behavior of the u(, t) state and estimation error (a) Closed-loop evolution of the PDE state u(, t) (b) Closed-loop evolution of the estimation error ũ(, t) G Application Eample In this section, we choose the following simple eample to illustrate simultaneous control and observation of a first-order hyperbolic system with a Fredholm integral (with discontinuous kernel) This is, we use the observer and control design to build an output-feedback controller that drives the system to the origin in finite time (eual to the sum of the time reuired for the observer convergence and for closed-loop state convergence) Consider (4) with f()=, g(, y)=6( y) and h(, y)= 6( y) The control U(t) is chosen as in Proposition 8 and the observer gain is in turn chosen as in Proposition 16 Both the open-loop system (4), (5) (with U(t) =) and (open-loop) error system (86), (87) are unstable Fig 4(a) and (b) show the obtained control and observer gains for this system Fig 5(a) shows the resulting state evolution (as epected, it converges in finite time) Fig 5(b) shows the evolution of the state estimation (finite-time stable) Since the state estimation converges in 1 second and, assuming full state measurements, it takes 1 second for the controller to steer the system to the origin, using the controller and observer in the same system ensures convergence in 2 seconds IV CONCLUSION In this article, we propose an integral transform that allows the construction of stabilizing boundary controllers for a class

13 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2157 of first-order hyperbolic PIDEs with Fredholm integrals Sufficient conditions for this stabilizing controller and transform are given in terms of the spectrum of two integral operators on Banach spaces and (in a more conservative form) in terms of the magnitudes of the coefficients of (4) Also, an eplicit transform and controller are given for some systems that verify additional assumptions on the shape of their coefficients Finally, analogous conditions for the observer design are presented This approach seems promising to deal with fully interconnected and underactuated PDE-PDE and PDE-ODE systems, as well as systems where non-local terms appear in the evolution euation Some research directions for future work are finding conditions that guarantee well-posedness of the kernel euations when the integral operators are not contractions (and the use of other solution methods for these cases) as well as etension of these methods to other classes of PDEs APPENDIX A Proof of Proposition 1 Proof (Proposition 1): This proof follows a similar approach to that used in standard backstepping to find sufficient conditions for the direct transform to eist Substituting (1) into (13), and after some computations (involving integration by parts, change of order of integration and using the value of u(1,t) from (12)) we obtain u(,t) f() u(y, t) u(y, t) p(, y)f(y)dy g(, y)p y (, y) y (, y)f(y)dyp(, ) g(s, y)p(, s)ds g(s, y)(, s)dsp (, y) h(, y) y (, y) y y y g(s, y)(, s)ds h(s, y)(, s)ds (, y) h(s, y)p(, s)ds dy h(s, y)p(, s) dy u(y, t)(, 1)p(1,y) dy = (138) We therefore focus on solving the set of coupled hyperbolic PIDEs (15), (16) with boundary conditions p(, ) = f() p(, y)f(y)dy (, y)f(y)dy,, 1 (139) (, 1) =,, 1 (14) which cancel the nonlinear term in the domain Consider the (invertible) change of variables φ :, 1 2, 2 1, 1 defined as and φ(, y) =( y, y),, y, 1 (141) P (φ(, y)) = P (φ 1 (, y),φ 2 (, y)) = p(, y),, y, 1 st y (142) Q (φ(, y)) = Q (φ 1 (, y),φ 2 (, y)) = (, y),, y, 1 st y (143) where φ i (, y) denotes the i-th component of φ(, y) Defining new variables ξ, 2 η 1, 1 we may rewrite (15), (16) and the boundary conditions (139), (14) as 2P ξ (ξ,η) ( ξ η = g 2, ξ η ) 2 ξη 2 ξη 2 ξη 2 ξη 2 ( h s, ξ η ) ( ξ η P s, ξ η ) s ds ( g s, ξ η ) ( ξ η P s, ξ η ) s ds ( g s, ξ η ) ( ξ η Q s, ξ η ) s ds, (ξ,η), 2, 1 st η min{ξ,2 ξ}, η ξ (144) 2Q ξ (ξ,η) ( ξ η = h 2, ξ η ) 2 ξη 2 ξη 2 ξη 2 ξη 2 ( h s, ξ η ) ( ξ η Q s, ξ η ) s ds ( g s, ξ η ) ( ξ η Q s, ξ η ) s ds ( h s, ξ η ) ( ξ η P s, ξ η ) s ds, (ξ,η), 2, 1 st η ma{ξ,2ξ}, η ξ 2 (145)

14 2158 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST 215 P (η, η) = f(η) η η P (η s, η s)ds Q(η s, η s)f(s)ds, η, 1 (146) Q(2 η, η) =, η 1, (147) Integrating (144) (wrt ξ from η to ξ with boundary condition (146)) and (145) (wrt ξ from ξ to 2η with boundary condition (147)) we obtain the following system of coupled integral euations (after inverting the change of variables and adjusting the limits of integration): p(, y) = y f(s)p( y, s)ds y σ y y 1y h(s, σ)p(σ y, s)ds dσ y 1σy g(s σ, σ)p(σ y, σ s)ds dσ f( y s)( y, y s)ds g(σ y s, σ) (σ y, σ y s)ds dσ f( y) (, y) = y 1y g(σ y, σ)dσ, σ 1y y, y, 1 st y h(s, σ y)p(σ, s)ds dσ h(s σ, σ y) (σ, σ s)ds dσ 1y 1σy g(s σ y, σ y) (σ, σ y s)ds dσ 1y h(σ, σ y)dσ, (148), y, 1 st y (149) The condition of the Proposition guarantees a uniue solution to the direct transform kernel integral euations and therefore, a suitable direct transform eists This ends the proof of Proposition 1 We should note that (144), (145) imply that the derivative of the direct transform kernels along the level curves of y (ie, in the ξ direction) is continuous B Proof of Theorem 11 Proof (Theorem 11): We will proceed by finding a change of variables w(, t) = u(, t) f 1 e λ k 1 (y)u(y, t)dy (15) that transforms system (74), (75) into the (finite-time stable) target system w t (, t) =w (, t), (, t) (, 1) (,T (151) with boundary condition for all t (,T: w(1,t)= (152) The assumption in the Theorem can be shown to imply that 1 k 1 (y)f 1 e λy dy (153) which, in turn, implies that the transformation (15) is boundedly invertible, with inverse given by u(, t) =w(, t)f 1 e λ where 1 () is defined as 1 () = 1 k 1 (y)f 1 e λy dy 1 (y)w(y, t)dy (154) 1 k 1 () (155) The proof then follows the classical backstepping paradigm of guaranteeing the stability of the closed-loop system by simultaneously finding a bounded (and boundedly invertible) transform and an associated control law that map the closedloop system into a target stable system The boundedness of both transforms guarantees, first, that a bounded initial condition in the original system is mapped to a bounded initial state for the target system and, second, that as the norm of the state of the target system goes to zero, the norm of the state in the original system also goes to zero Differentiating (15) with respect to, we obtain w (, t) =u (, t) λf 1 e λ net, differentiating (15) with respect to t w t (, t) =u t (, t) f 1 e λ k 1 (y)u(y, t)dy (156) k 1 (y)u t (y, t)dy (157)

15 BRIBIESCA-ARGOMEDO AND KRSTIC: BACKSTEPPING-FORWARDING CONTROL AND OBSERVATION FOR HYPERBOLIC PDEs 2159 Plugging (74) into (157) and integrating by parts the term containing the spatial derivative of u we obtain w t (, t) =u (, t)f 1 e λ h 1 (y)u(y, t)dy f 1 e λ k 1 (1)u(1,t)f 1 e λ k 1 ()u(,t) f 1 e λ f 1 e λ k 1(y)u(y, t)dy k 1 (y)f 1 e λy h 1 (s)u(s, t)ds dy Evaluating (15) at =1we obtain the condition which in turn implies u(1,t)=u(t) =f 1 e λ w t (, t) =u (, t)f 1 e λ f 1 e λ k 1 (1)f 1 e λ (158) k 1 (y)u(y, t)dy (159) h 1 (y)u(y, t)dy f 1 e λ k 1 ()u(,t)f 1 e λ f 1 e λ k 1 (y)f 1 e λy k 1 (y)u(y, t)dy k 1(y)u(y, t)dy h 1 (s)u(s, t)ds dy (16) Substituting (156) and (16) into (74) and changing the order of integration in the resulting double integral we get f 1 e λ λ k 1 (y)u(y, t)dy h 1 (y)u(y, t)dy k 1 (1)f 1 e λ k()u(,t) k 1 (y)u(y, t)dy h 1 (y)u(y, t) k 1(y)u(y, t)dy k 1 (s)f(s)ds dy = (161) A sufficient condition for this euation to hold is that the following integro-differential euation is verified: k 1(y) λ k 1 (1)f 1 e λ k 1 (y) = h 1 (y) 1 with boundary condition and Defining k 1 (s)f 1 e λs ds (162) k 1 () = (163) α 2 = λ k1 (1)f 1 e λ (164) 1 g 1 =1 k 1 (s)f 1 e λs ds (165) (162) can be solved as a nonhomogeneous first-order ODE with source term g 1 h 1 (y) since g 1 is different from zero, as stated in (153) to obtain k 1 (y) =g 1 y e α 2(ys) h 1 (s)ds (166) Multiplying both sides of the euation by f 1 e λy, integrating from to 1, using the definition of g 1 and Assumption 2 we obtain g 1 = 1 1 h 1(s) s eα 2(ys) f 1 e λy dy ds (167) which implies y k 1 (y) = eα2(ys) h 1 (s)ds 1 h 1(s) s eα 2(ys) f 1 e λy dy ds (168) The definition of α 2 in this proof can be shown to be euivalent to the epression for α given in Assumption 2 in terms of only the coefficients of the euation This can be seen by multiplying (162) by f 1 e λy on both sides and integrating from to 1, integrating by parts the term containing the derivative of k 1 and using Assumption 2 This completes the proof of Theorem 11 REFERENCES 1 A Smyshlyaev and M Krstic, Closed-form boundary state feedback for a class of 1-D partial integro-differential euations, IEEE Trans Autom Control, vol 49, no 12, pp , 24 2 M Krstic and A Smyshlyaev, Backstepping boundary control for firstorder hyperbolic PDEs and application to systems with actutator and sensor delays, Syst Control Lett, vol 57, no 9, pp , 28 3 M Krstic and A Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, 28, ser Advances in design and control 4 T Meurer and A Kugi, Tracking control for boundary controlled parabolic pdes with varying parameters: Combining backstepping and differential flatness, Automatica, vol 45, no 5, pp , 29

16 216 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 6, NO 8, AUGUST H Sano and S Nakagiri, Backstepping boundary control of first-order coupled hyperbolic partial integro-differential euations, in Proc 14th WSEAS Int Conf Appl Math, Tenerife, Spain, Dec 29, pp R Vazuez and M Krstic, Boundary observer for output-feedback stabilization of thermal-fluid convection loop, IEEE Trans Control Syst Technol, vol 18, no 4, pp , 21 7 D Tsubakino and S Hara, Backstepping observer using weighted spatial average for 1-dimensional parabolic distributed parameter systems, in Proc 18th IFAC World Congress, Milano, Italy, 211, pp D Bresch-Pietri, J Chauvin, and N Petit, Adaptive control scheme for uncertain time-delay systems, Automatica, vol48,no8,pp , S Nakagiri, Deformation formulas and boundary control problems of first-order volterra integro-differential euations with nonlocal boundary conditions, IMA J Math Control Inform, vol 3, no 3, pp , T Meurer, Control of Higher-Dimensional PDEs: Flatness and Backstepping Designs New York, NY, USA: Springer-Verlag, 213, ser Communications and Control Engineering 11 R Vazuez and M Krstic, Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation Boston, MA, USA: Birkhäuser, 28, ser Systems & Control: Foundations & Applications 12 E Cerpa and J Coron, Rapid stabilization for a korteweg-de vries euation from the left dirichlet boundary condition, IEEE Trans Autom Control, vol 58, no 7, pp , H Sano, Output tracking control of a parallel-flow heat echange process, Syst Control Lett, vol 6, no 11, pp , M Krstic, Delay Compensation for Nonlinear, Adaptive and PDE Systems Boston, MA, USA: Birkhäuser, 29, ser Systems & Control: Foundations & Applications 15 S J Moura, N Chaturvedi, and M Krstic, Adaptive PDE Observer for Battery SOC/SOH Estimation, in Proc ASME Dynam Syst Control Conf, Ft Lauderdale, FL, USA, N Bekiaris-Liberis and M Krstic, Compensating the distributed effect of a wave PDE in the actuation or sensing path of MIMO LTI systems, Syst Control Lett, vol 59, no 11, pp , Nov N Bekiaris-Liberis and M Krstic, Lyapunov stability of linear predictor feedback for distributed input delays, IEEE Trans Autom Control, vol 56, no 3, pp , Mar C Guo, C Xie, and C Zhou, Stabilization of a spatially non-causal reaction-diffusion euation by boundary control, Int J Robust Nonlin Control, vol 24, no 1, pp 1 17, F Bribiesca Argomedo and M Krstic, Backstepping-forwarding boundary control design for first-order hyperbolic systems with fredholm integrals, in Proc Amer Control Conf, Portland, OR, USA, 214, pp E Davies, Linear Operators and Their Spectra London,UK: Cambridge Univ Press, 27, ser Cambride Studies in Advanced Mathematics 21 M A Khamsi and W A Kirk, An Introduction to Metric Spaces and Fied Point Theory New York, NY, USA: Wiley, 21, ser Pure and Applied Mathematics Federico Bribiesca-Argomedo (M 13) was born in Zamora, Michoacán, Meico in 1987 He received the BSc degree in mechatronics engineering from the Tecnológico de Monterrey, Monterrey, Meico in 29, the MSc degree in control systems from Grenoble INP, Grenoble, France, in 29, and the PhD degree in control systems from Grenoble University, GIPSA-Lab, Grenoble, France He was a postdoc in the Department of Mechanical and Aerospace Engineering, University of California, San Diego and is now an Assistant Professor in the Department of Mechanical Engineering and Design, INSA of Lyon, Ampère Lab, Lyon, France His research interests include control of partial differential euations and nonlinear control theory In particular, he has applied these techniues to safety factor profile control in tokamak plasmas Miroslav Krstic (F 2) holds the Alspach Endowed chair and is the founding director of the Cymer Center for Control Systems and Dynamics at UC San Diego He also serves as Associate Vice Chancellor for Research at UCSD He has coauthored ten books on adaptive, nonlinear, and stochastic control, etremum seeking, control of PDE systems including turbulent flows, and control of delay systems Dr Krstic is a Fellow of IFAC, ASME, and IET (UK), and a Distinguished Visiting Fellow of the Royal Academy of Engineering He received the UC Santa Barbara best dissertation award and student best paper awards at CDC and ACC, the PECASE, NSF Career, and ONR Young Investigator awards, the Aelby and Schuck paper prizes, the Chestnut tetbook prize, and the first UCSD Research Award given to an engineer He has held the Springer Visiting Professorship at UC Berkeley He serves as Senior Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and Automatica, as Editor of two Springer book series, and has served as Vice President for Technical Activities of the IEEE Control Systems Society and as Chair of the IEEE CSS Fellow Committee