Bilinear spatial control of the velocity term in a Kirchhoff plate equation

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1 Electronic Journal of Differential Equations, Vol. 1(1), No. 7, pp ISSN: URL: or ftp ejde.math.swt.edu (login: ftp) Bilinear spatial control of the velocity term in a Kirchhoff plate equation Mary Elizabeth Bradley & Suzanne Lenhart Abstract We consider a bilinear optimal control problem with the state equation being a Kirchhoff plate equation. The control is a function of the spatial variables and acts as a multiplier of the velocity term. The unique optimal control, driving the state solution close to a desired evolution function, is characterized in terms of the solution of the optimality system. 1 Introduction We consider the problem of controlling the solution of a Kirchhoff plate equation. The motion with appropriate boundary conditions describes the motion of a thin plate which is clamped along one portion of its boundary and has free vibrations on the other portion of the boundary. We consider bilinear optimal control, acting as a multiplier of a velocity term, is a function of the spatial variables x and y. Given control h U M = {h L (); M h(x, y) M}, the displacement solution w = w(h) of our state equation satisfies w tt + w = h(x, y)w t on Q = (, T ) w(x, y, ) = w 1 (x, y), w t (x, y, ) = w (x, y) w = w ν = on Σ = Γ (, T ), (1.1) w + (1 µ)b 1 w = on Σ 1 = Γ 1 (, T ), w ν + (1 µ)b w = on Σ 1 = Γ 1 (, T ), Mathematics Subject Classifications: 49K, 35F1. Key words: Kirchhoff plate, optimal control, bilinear control. c 1 Southwest Texas State University. Submitted July 6,. Published May 1, 1. 1

2 Bilinear spatial control EJDE 1/7 where R with C boundary, = Γ Γ 1, Γ Γ 1 =, Γ, ν = n 1, n is the outward unit normal vector on, and B 1 w = n 1 n w xy n 1w yy n w xx, B w = [ (n τ 1 n )w xy + n 1 n (w yy w xx ) ]. The direction τ in B w is the tangential direction along Γ 1. The plate is clamped along Γ and has free vibrations along Γ 1. The constant µ, < µ < 1, represents Poisson s ratio. We take as our objective functional J(h) = 1 ( (w z) dq + β Q ) h (x, y) d where z is the desired evolution for the plate and the quadratic term in h represents the cost of implementing the control with weighting factor β >. For convenience, we assume that z C([, T ]; L ()), z t C([, T ]; L ()). We seek to minimize the objective functional, i.e., characterize an optimal control h U M such that J(h ) = min h U M J(h). For background on plate models and control, see the books by Lagnese and Lions [15], Lagnese [13], Lagnese, Leugering, and Schmidt [14], Kormornik [11], Li and Yong [18], and Lions [19]. The bilinear control case treated here does not fit into the Riccati framework [17]; even though the objective functional is quadratic, the state equation has a bilinear term, hw t. See [4, 6, 8, 9, 1, 1, 16] for control papers involving Kirchhoff plates. Bilinear control problems similar to the problem here were introduced in three papers by Ball, Marsden, and Slemrod [1-3], and in Bradley and Lenhart [5] (with control acting through the term hw). Note that in a recent paper by Bradley, Lenhart and Yong, the case of h(t)w t was treated [7]. In section, we show well-posedness of our state problem. In section 3, we show the existence of an optimal control by a minimizing sequence argument. In section 4, we derive a characterization for optimal controls, in terms of the solutions of an optimality system. The optimality system consists of the state equation coupled with an adjoint equation, and it is derived by differentiating the objective functional and the map h w(h) with respect to the control. In section 5, we prove that the optimal control is unique for small time, T, provided that initial data are taken to be sufficiently smooth.

3 EJDE 1/7 M. E. Bradley & S.Lenhart 3 Well-posedness of the State Equation We will begin by proving existence, uniqueness, and regularity results for the state equation. We first define our solution spaces: H Γ () = { w H () w = w } ν = on Γ and H = H Γ () L (). Note that the bilinear form on H Γ (), a(u, v) = { w v + (1 µ)(w xy v xy w xx v yy w yy w xx } d induces a norm on H Γ () which is equivalent to the usual H norm on H Γ () (see [1]). Definition Given h U M, w = w(h) = (w(h), w t (h)) is a weak solution to (1.1) if w C([, T ]; H), w() = (w 1, w ) and w satisfies T T w tt, φ dt + a(w, φ)(t)dt = hw t φ d dt Q for all φ H Γ (), and, denotes the duality pairing between [H Γ ()] and H Γ (). Theorem.1 (i) Let w() = (w 1, w ) H and h U M. Then the system (1.1) has a unique weak solution w = w(h) = (w, w t ). (ii) In addition, if (w 1, w ) D where D = { (w 1, w ) (H 4 () H Γ ()) H Γ () : w 1 + (1 µ)b 1 w 1 = on Γ 1, w } 1 + (1 µ)b w 1 = on Γ 1 ν for h U M, then the weak solution satisfies w C([, T ]; (H 4 () H Γ ()) H Γ ()) and w tt C([, T ]; L ()). Furthermore, (1.1) holds in the L sense.

4 4 Bilinear spatial control EJDE 1/7 Proof. (i) To write the system in semigroup form, we define the operator A: Aw = w with domain { D(A) = w H 4 () HΓ () : w + (1 µ)b 1 w = on Γ 1 and w } ν + (1 µ)b w = on Γ 1. Then define operator A : H 4 () HΓ () H by [ ] I A w = w with D(A) = D A. Then the stated equation (1.1) can be written as d dt w(t) = A w(t) + B w(t) [ ] w1 w() = w = w [ ] with B w(t) =. Using skew-adjointness, the operator A generates a hw t (t) strongly continuous unitary group on H. Since B is a bounded perturbation of A on H, by standard semigroup theory [], we have the conclusion of (i). (ii) Assume that w D and h U M. From variation of parameters [] and (i), w(t) = e At w + t e A(t τ) B( w)(τ)dτ, (.1) where e At represents the semigroup generated by A. Proceeding to formally differentiate (.1) in the t variable and defining a new variable ṽ = (v 1, v ) = d w dt, we seek a solution of the form: Setting ṽ(t) = Ae At w + B w(t) + F ṽ = Ae At w + B w(t) + t t Ae A(t τ) B w(τ)dτ. Ae A(t τ) B w(τ)dτ, (.) we seek a fixed point of F, i.e. we seek a unique point ṽ C([, T ]; H) such that F ṽ = ṽ. Note that t Ae A(t τ) B w(t)dt = t d ( ) e A(t τ) B w(τ) dτ + dτ = B w(t) + e At [ hw ] + t t e A(t τ) d dτ B w(τ)dτ e A(t τ) [ hw ττ ] dτ,

5 EJDE 1/7 M. E. Bradley & S.Lenhart 5 where w = w t (x, y, ). Thus from (.), F can be rewritten as [ ] t [ ] F (ṽ) = Ae At w + e At + e A(t τ) dτ. hw hv Since w D(A) and h = h(x, y) U M L (), F : C([, T ]; H) C([, T ]; H) is bounded. We now verify that F is a contraction on C([, T ]; H) for small T for t T, t [ ] F ṽ 1 F ṽ C([,T];H) e A(t τ) dτ h(v 1 v ) sup t T t h(v 1 v )(τ) L ()dτ T M ṽ 1 ṽ C([,T];H). C([,T];H) Taking T < 1 M, we have the F is a contractive mapping on C([, T ], H). To complete the proof, we set ṽ(t ) (with ṽ being the fixed point) as the new initial data and repeat the argument to obtain F as a contraction on C([T, T ], H). Repeating this procedure yields the result on [, T ]. We observe first that (w t, w tt ) C([, T ]; H), and then hw t L (Q) with equation (1.1), gives w C([, T ]; L ()). By standard elliptic theory, w C([, T ]; H 4 () H Γ ()). We now present an a priori estimate needed for the existence of an optimal control. Lemma.1 (A priori estimate) Given w() H and h U M, the weak solution to (1.1) satisfies Proof. that w C([,T ];H) (1 + MT ) 1/ e CMT w() H. (.3) Since D is dense in H, there exists a sequence, { w() n } in D, such w() n w() strongly in H. Denoting by w n the solution of (1.1) with initial data w() n and control h, then w n has the additional regularity from Theorem.1(ii). Using wt n as a multiplier in (1.1), we obtain s = (wttw n t n + w n wt n h(wt n ) ) d dt s 1 d s = dt (wn t ) 1 d s d dt + dt a(wn, w n )dt h(wt n ) d dt.

6 6 Bilinear spatial control EJDE 1/7 Consequently, we have 1 (wt n ) (x, y, s) d + 1 a(wn, w n )(s) = 1 wn L () + 1 a(wn 1, w n 1) + 1 w()n H + M Gronwall s Inequality implies { sup s T s s w n (t) Hdt. (wt n ) (x, y, t) d + a(w n, w n )(s) h(w n t ) d dt } w() n H(1 + MT )e CMT, (.4) which gives the desired result for smooth approximations. Now we can pass to the limit and obtain (.3) for w. 3 Existence of Optimal Controls We now prove the existence of an optimal control by a minimizing sequence argument. Theorem 3.1 There exists an optimal control h U M, which minimizes the objective functional J(h) over h in U M. Proof. Let {h n } be a minimizing sequence in U M, i.e., By Lemma.1, for w n = w(h n ), On a subsequence, we have w n w lim n J(hn ) = inf J(h). h U M w n C([,T ],H) C 1 e CMT. weakly* in L ([, T ]; H Γ ()), w n t w t weakly* in L ([, T ]; L ()), w n tt w tt weakly* in L ([, T ]; (H Γ ()) ), h n h weakly in L (). The convergence of the wtt n sequence follows from the PDE (1.1) and the estimate from Lemma.1 In weak form, w n satisfies T [ wtt, n φ + a(w n, φ)(t)]dt = h n wt n φdq, (3.1) Q

7 EJDE 1/7 M. E. Bradley & S.Lenhart 7 where we now allow that φ = φ(x, y, t) in L ([, T ], H) and φ t in L (Q). In the convergence as n, the only difficult term is on the RHS of (3.1). Examining the RHS we see that T RHS = T = + h n (x, y)w n t (x, y, t)φ(x, y, t) d dt h n (x, y)w n (x, y, t)φ t (x, y, t) d dt h n (x, y) (w n (x, y, T )φ(x, y, T ) w 1 φ(x, y, )) d, where we have used the fact that w n 1 = w 1 for all n. Now by a standard result from semigroup theory (see, for example []), we know that w n (x, y, t) C([, T ], H Γ ), which implies that w n (x, y, T ) H Γ () C(), since R. As a consequence, we may pass with a limit on this equation to obtain T RHS h (x, y)w (x, y, t)φ t (x, y, t) d dt + h (x, y) (w (x, y, T )φ(x, y, T ) w 1 φ(x, y, )) d as n. From this we obtain w = w(h ), which is the weak solution of (1.1) with control h. Since the objective functional is lower semi-continuous with respect to weak convergence, we obtain and h is an optimal control. J(h ) = inf J(h) h U M 4 Necessary Conditions We now derive necessary conditions that any optimal control must satisfy. To derive these necessary conditions, we must differentiate our functional J(h) and w = w(h) with respect to h. The differentiation of J results in a characterization of optimal controls in terms of the optimality system. Lemma 4.1 The mapping is differentiable in the following sense: h U M w(h) C([, T ]; H) w(h + εl) w(h) ε ψ weakly* in L ([, T ]; H),

8 8 Bilinear spatial control EJDE 1/7 as ε, for any h, h + εl U M. Moreover, the limit ψ = (ψ, ψ t ) is a weak solution to the following system ψ tt + ψ hψ t = lw t in Q ψ(x, y, ) = ψ t (x, y, ) = in ψ = ψ ν = on Σ (4.1) ψ + (1 µ)b 1 ψ = on Σ 1 ψ ν + (1 µ)b ψ = on Σ 1. Proof. Denote by w ε = w(h + εl) and w = w(h). By (1.1), ( w ε w)/ε is a weak solution of ( w ε ) ( w w + ε ) ( w w ε ) w = h + lwt ε in Q ε ε ε tt with homogeneous initial and boundary conditions. Using the proof of Lemma.1 with source term lwt ε, we obtain w ε w ε lwt ε L (Q) ecmt. C([,T ];H) But we have a priori estimates on w ε t, lw ε t L (Q) T l w ε C(,T ;H) (1 + MT ) 1/ e CMT w() H, using Lemma.1 on w ε. Hence on a subsequence, as ε, w ε w ε ψ weakly* in L ([, T ]; H). Similar to the proof of Theorem 3.1, we obtain that ψ is a weak solution of (4.1). We obtain the existence of an adjoint solution and use it in the differentiation of the map h J(h) to obtain our characterization of an optimal control. Theorem 4.1 Given an optimal control h in U M and corresponding state solution w = w(h ) to (1.1), there exists a unique weak solution to the adjoint problem: p = (p, p t ) C([, T ]; H) p tt + p + hp t = w z in Q p = p ν = on Σ (4.) p + (1 µ)b 1 p = on Σ 1 p ν + (1 µ)b p = on Σ 1 p(x, y, T ) = p t (x, y, T ) = (transversality condition). t

9 EJDE 1/7 M. E. Bradley & S.Lenhart 9 Furthermore ( ( h (x, y) = max M, min 1 T )) wt p(x, y, t)dt, M. (4.3) β Proof. The proof of existence of the solution to the adjoint equation is similar to the proof of existence of solution of the state equation since the source term (w z) C([, T ], L ()). However, since p(x, y, T ) =, there is a difference in the constant in the a priori estimate: { } sup (p n t ) (x, y, t) d + a(p n, p n )(s) s T w z C([,T ],L ()) (1 + MT )e CMT. We now proceed to characterize the optimal control in terms of the state w = (w, w t ) and and adjoint p = (p, p t ). Let h + εl be another control in U M and w ε = w(h + εl) be the corresponding solution to the state equation. Then since J achieves its minimum at h, we have J(h + εl) J(h ) lim ε + ε ( w ε w ) ( w ε + w ) z = lim dq + β (lh + εl ) d ε + Q ε = ψ(w z)dq + β h l d. Q Substituting in from the adjoint equation (4.) for w z and then using ψ PDE (4.1), we obtain = T T = T ψ, p tt dt + ψ tt, p dt + ( l βh + T T a(ψ, p)dt + a(ψ, p)dt ) (wt p)dt d. Q Q ψh p t dq + β ψ t h pdq + β h l d h l d Using a standard control argument based on the choices for the variation l(x, y), we obtain the desired characterization for h : ( ( h (x, y) = max M, min 1 T )) wt p(x, y, t)dt, M. β

10 1 Bilinear spatial control EJDE 1/7 5 Uniqueness of the Optimal Control We now characterize the optimal control as the unique solution to the optimality system ( ( w tt + w = max M, min 1 T )) wt p(x, y, s)ds, M w t β in Q = (, T ) ( ( p tt + p = max M, min 1 T wt p(x, y, s)ds, M β w = p = w ν = p )) p t + w z in Q ν = onσ = Γ (, T ), (OS) w + (1 µ)b 1 w = p + (1 µ)b 1 p = on Σ 1 = Γ 1 (, T ) w ν + (1 µ)b w = p ν + (1 µ)b p = on Σ 1 = Γ 1 (, T ) w(x, y, ) = w 1 (x, y), w t (x, y, ) = w (x, y) on, p(x, y, T ) = p t (x, y, T ) = on T. The existence of solutions to (OS) is given by Theorems.1 and 4.1. We now prove uniqueness, provided that time T is sufficiently small and that the conditions on initial data are as in Theorem.1(ii). Theorem 5.1 The solution to the optimality system, (OS), is unique for T sufficiently small, optimal control h U M and initial data such that (w 1, w ) D, as in Theorem.1(ii). Proof. Since z t C([, T ]; L ()), an extension of Theorem.1(ii) gives p t C([, T ]; H ()). Also, Theorem.1(ii) directly implies w t C([, T ]; H Γ ()). Consequently, we have that w, p, w t, and p t are all bounded functions over Q. Suppose we have two weak solutions corresponding to two optimal controls, h and h: w = (w, w t ), p = (p, p t ), ŵ = (w, w t ), ˆp = (p, p t ). We then have that ( w ŵ) and ( p ˆp) are weak solutions to the following system of equations (w w) tt + (w w) = hw t hw t in Q (p p) tt (p p) = hp t hp t + (w w) in Q (w w)(t = ) = ; (w w) t (t = ) = in {} (p p)(t = T ) = ; (p p) t (t = T ) = in {T },

11 EJDE 1/7 M. E. Bradley & S.Lenhart 11 where we denote ( ( T h = max M, min ( ( T h = max M, min )) w t pdt, M, (5.1.a) )) w t pdt, M. (5.1.b) Also, we have homogeneous boundary conditions as in (OS). (Here, we have multiplied through the p-equation by -1.) Multiplying the w-equation by (w w) t (resp. the p-equation by (p p) t ) and integrating by parts over [, t] (resp. over [t, T ]) we obtain 1 [ ((w w)t ) (t) + ((p p) t ) (t) ] d + 1 a(w w, w w)(t) + 1 a(p p, p p)(t) (5.) t = (hw t hw t )(w w) t dτd + T t ((hp t hp t )(p p) t + (w w)(p p) t )dτd. To estimate the RHS of equation (5.), we note that Consequently, we have t (hw t hw t )(w w) t dτ d t C C t t hw t hw t = h(w w) t + w t (h h), hp t hp t = h(p p) t + p t (h h). [ h ((w w)t ) + h h w t (w w) t ] dτ d [ ((w w)t ) + h h (w w) t ] dτ d (h and w t are bounded) ((w w) t ) dτ d (5.3) t + C ( h h d ) 1/( ((w w) t ) dτd ) 1/ 3C t ((w w) t ) dτ d + C h h d. Similarly, for we have for the p-term T (hp t hp t )(p p) t dτ d 3C t T t ((p p) t ) dτd+ C (Hölder s in space) h h d. (5.4)

12 1 Bilinear spatial control EJDE 1/7 We now estimate the h-term using equations (5.1.a)-(5.1.b). h h d 1 β = 1 β T T (w t p w t p) dτ d [(w w) t p + (p p)w t dτ] d { [( β T (w w) t pdτ ) ( T + (p p)w t dτ ) } ] d { [ β T T T ((w w) t ) dτ p dτ + (p p) dτ T ( C ((w w)t ) + (p p) ) dτ d. T (5.5) ] } (w t ) dτ d In the previous to the last inequality we used Hölder s in time, and in the last inequality the boundedness of p and w t. Bounding the last term in equation (5.), we have T (w w)(p p) t dτd (5.6) t 1 T t (w w) dτd + Putting together equations (5.)-(5.6), 1 ( ((w w)t ) (t) + ((p p) t ) (t) ) d T t ((p p) t ) dτd. + 1 a(w w, w w)(t) + 1 a(p p, p p)(t) (5.7) { T T T C ((w w) t ) dτ + ((p p) t ) dτ ((w w) t ) dτ + T (p p) dτ + T (w w) dτ } d. Now taking a supremum in time on both sides of the inequality, we have that { 1 ( sup ((w w)t ) (t) + ((p p) t ) (t) ) d t T + 1 a(w w,w w)(t) + 1 } a(p p, p p)(t) sup CT t T [ ((w w)t ) (t) + ((p p) t ) (t) + (p p) (t) + (w w) (t) ] d.

13 EJDE 1/7 M. E. Bradley & S.Lenhart 13 Using Poincare s inequality, [ (w w) (t) + (p p) (t) ] d a(w w, w w)(t) + a(p p, p p)(t). Finally, by taking T such that CT < 1, we have that { ( sup ((w w)t ) (t) + ((p p) t ) (t) ) d t T } + a(w w, w w)(t) + a(p p, p p)(t), which implies (w w) = (p p) = (w w) t = (p p) t = = w = ŵ and p = ˆp. This completes present proof. Remark: In [7], we were able to solve the corresponding bilinear control problem, but with the controlled velocity coefficient, h(t), being a function of time only. A main difference between our result here and that result is in the proofs of the existence and the uniqueness of the optimal control. The case of the controlled velocity coefficient, h(x, y, t), is an open problem at this time. Acknowledgment The second author was partially supported by the National Science Foundation. References [1] J. M. Ball, J. E. Marsden, and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim. (198), [] J. M. Ball and M. Slemrod, Feedback stabilization of semilinear control systems, Appl. Math. Optim. 5 (1979), [3] J. M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Commun. Pure Appl. Math. 3 (1979), [4] M. E. Bradley, Global exponential decay rates for a Kirchhoff plate with boundary nonlinearities, Differential and Integral Equations 7 (1994), [5] M. E. Bradley and S. Lenhart, Bilinear optimal control of a Kirchhoff plate, Systems and Control Letters (1994), [6] M. E. Bradley and S. Lenhart, Bilinear optimal control of a Kirchhoff plate via internal controllers, in Modelling and Optimization of Distributed Parameter Systems: Applications to Engineering, Chapman and Hall, London, 1996, 33 4.

14 14 Bilinear spatial control EJDE 1/7 [7] M. E. Bradley, S. Lenhart and J. Yong, Bilinear optimal control of the velocity term in a Kirchhoff plate equation, J. Math. Anal. Apps., 83 (1999), [8] E. Hendrickson and I. Lasiecka, Boundary control problem for a dynamic Kirchhoff plate model with partial observation, in Modelling and Optimization of Distributed Parameter Systems: Applications to Engineering, Chapman and Hall, London, 1996, [9] M. A. Horn and I. Lasiecka, Asymptotic behavior with respect to thickness of boundary stabilizing feedback for the Kirchhoff plate, J.D.E. 114 (1994), [1] V. Komornik, On the boundary stabilization of plates, Metz Days, editors M. Chipot and J. Saint Jean Paulin, Pitman Research Notes in Math, 49(1991), , Essex, UK. [11] V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, J. Wiley & Sons and Masson, New York, [1] J. E. Lagnese, Uniform boundary stabilization of homogeneous isotropic plates, in Lecture Notes in Control and Information Sciences, Vol. 1, Springer, Berlin (1987), [13] J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, [14] J. E. Lagnese, G. Leugering, and E.J.P.G. Schmidt, Modeling, Analysis, and Control of Dynamic Elastic Multi-Link Structures, Birkhaüser, Boston, [15] J. E. Lagnese and J. L. Lions, Modelling Analysis and Control of Thin Plates, Masson, Paris, [16] J. Lasiecka and R. Triggiani, Exact Controllability and uniform stabilization of Kirchhoff plates with boundary control only on w Σ and homogeneous boundary displacement, J. Differential Equations 93(1991), [17] I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, Vol. 164, Springer, New York, [18] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhaüser, Boston, [19] J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer, Berlin, [] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

15 EJDE 1/7 M. E. Bradley & S.Lenhart 15 [1] H. L. Royden, Real Analysis, Macmillan, New York, Mary Elizabeth Bradley University of Louisville, Department of Mathematics Louisville, KY 49 USA Suzanne Lenhart University of Tennessee, Department of Mathematics Knoxville, TN USA