Numerical schemes for nonlinear predictor feedback

Transcription

1 Math. Control Signals Syst : DOI 0.007/s ORIGINAL ARTICLE Numerical schemes for nonlinear predictor feedback Iasson Karafyllis Miroslav Krstic Received: 3 October 202 / Accepted: 9 February 204 / Published online: 4 March 204 Springer-Verlag London 204 Abstract This paper focuses on a specific aspect of the implementation problem for predictor-based feedback laws: the problem of the approximation of the predictor mapping. It is shown that the numerical approximation of the predictor mapping by means of a numerical scheme in conjunction with a hybrid feedback law that uses sampled measurements can be used for the global stabilization of all forward complete nonlinear systems that are globally asymptotically stabilizable and locally exponentially stabilizable in the delay-free case. Explicit formulae are provided for the estimation of the parameters of the resulting hybrid control scheme. Keywords methods Nonlinear systems Delay systems Feedback stabilization Numerical Introduction Feedback laws with distributed delays arise when predictor-based methodologies are applied to systems with input or measurement delays. The pioneering works [2,4,5] on predictor feedback were applied to linear systems. The recent works [6,9,0,2, 3] have extended the predictor-based methodologies to nonlinear systems and timevarying delays. I. Karafyllis B Department of Mathematics, National Technical University of Athens, Zografou Campus, 5780 Athens, Greece iasonkar@central.ntua.gr M. Krstic Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA , USA krstic@ucsd.edu

2 520 I. Karafyllis, M. Krstic This paper focuses on a specific aspect of the implementation problem for predictorbased feedback laws: the problem of the approximation of the predictor mapping. This problem is common for nonlinear systems: for nonlinear systems it is very rare that the solution map is known analytically. The recent work [6] was devoted to the approximation of the predictor mapping with a successive approximation approach: the method is suitable for globally Lipschitz systems. The problem of approximation of the predictor mapping is an important aspect of the implementation problem for predictorbased feedback laws but it is different from the usual problem of the approximation of distributed delays by discrete delays. The latter problem will not be studied in the present work. The idea of the numerical approximation of the predictor mapping by means of a numerical scheme for solving ordinary differential equations arises naturally as a possible method for solving the problem of approximation of the predictor mapping. However, certain obstructions exist, which are not encountered in standard numerical analysis results. The first obstruction is the existence of inputs: control theory tackles systems with inputs control systems, whereas standard results in numerical analysis are dealing with dynamical systems systems without inputs. Exception is the work [3] see also references therein. A second problem is the scarcity of explicit formulae for the approximation error which coincides with the so-called global discretization error in numerical analysis: in most cases the estimates of the approximation error are qualitative see [4,5]. In this work, we show that the numerical approximation of the predictor mapping by means of a numerical scheme for solving ordinary differential equations is indeed one methodology that can be used with success for systems which are not globally Lipschitz. More specifically, we focus on the explicit Euler scheme. We also study the sampling problem, i.e., the problem where the measurement is not available online but it is available at discrete time instants. The problem is solved by means of a hybrid feedback law and the main result is given next. Theorem. Consider the delay-free system: ẋt = f xt, ut xt R n, ut R m. where f :R n R m R n is a continuously differentiable mapping with f 0, 0 = 0. Assume that: A System. is forward complete. A2 There exists a continuously differentiable function k C R n ;R m with k0 = 0 such that 0 R n is a Globally Asymptotically Stable and Locally Exponentially Stable equilibrium point of the closed-loop system. with ut = kxt. Then for every τ>0, r > 0 there exists a locally bounded mapping N :R + {, 2, 3,...}, a constant ω > 0 and a locally Lipschitz, non-decreasing function C :R + R + with C0 = 0, such that for every partition {T i } i=0 of R + with

3 Numerical schemes for nonlinear predictor feedback 52 sup i 0 T i+ T i r, for every x 0 R n and u 0 L [ τ,0;r m, the solution xt, ut R n R m of the closed-loop system with ẋt = f xt, ut τ xt R n, ut R m.2 żt = f zt, kzt, zt R n, for t [T i, T i+ ut = kzt.3 and where N := N xt i +sup Ti τ s<t i us, h = τ N and z j+ = z j + j+h jh zt i = z N.4 f z j, ut i τ +s ds, for j = 0,...,N and z 0 = xt i.5 and initial condition x0 = x 0 and us = u 0 s for s [ τ,0 satisfies the following inequality for all t 0: xt + sup t τ s<t us exp ω tc x 0 + sup τ s<0 u 0 s.6 The notions of Global Asymptotic Stability and Local Exponential stability employed in the statement of Theorem. are the standard notions used in the literature see []. The notion of forward completeness for. is the standard notion that guarantees existence of the solution of. for all times, all initial conditions and all possible inputs see []. Notice that.5 is the application of the explicit Euler numerical scheme to the control system.2 with step size h = N τ. Since the number of the grid points N := N xt i +sup Ti τ s<t i us is a function of the state and the input, it is clear that different time steps are used each time that a new measurement arrives. Theorem. is proved by means of a combined Lyapunov and small-gain methodology and its proof is constructive. In Sect. 3, the control practitioner will find explicit formulae for the computation of N := N xt i +sup Ti τ s<t i us, which requires the knowledge of an appropriate Lyapunov function for the closed-loop system. with ut = kxt. Notice that the fact that the function C :R + R + involved in.6 is locally Lipschitz with C0 = 0 guarantees the analogue of local exponential stability for complicated systems such as the closed-loop system.2 with.3,.4 and.5 systems with delays and hybrid features, since the estimate

4 522 I. Karafyllis, M. Krstic xt + sup t τ s<t us exp ω t x 0 + sup u 0 s, for all t 0 τ s<0 holds for certain appropriate constant >0 and for initial conditions with x 0 + sup τ s<0 u 0 s sufficiently small. Therefore, both global asymptotic stability and local exponential stability are preserved, despite the delay, the sampled measurements, and the numerical approximation. Clearly, the proposed control scheme has a digital component Eqs..4,.5 and an analog component Eq..3. Therefore, the implementation of the analog component is an important issue. The analog component can be implemented with precision for feedback linearizable systems in strict feedback form, i.e., for systems of the form: ẋ i = f i x,...,x i + g i x,...,x i x i+ x n+ = u R i =,...,n.7 where f i : R i R, g i : R i R i =,...,n are smooth functions with f i 0 = 0 and g i x,...,x i >0 for all x R n and i =,...,n. Indeed, for the class of systems.7 and for every set of real numbers {a i R, i =,...,n}, we are in a position to construct explicitly a smooth function k C R n ;R with k0 = 0 and a global diffeomorphism C R n ;R n with 0 = 0 such that the dynamics of the closed-loop system.7 with u = kx expressed in the transformed coordinates ξ = x satisfy the linear differential equations ξ i = ξ i+ for i =,...,n and ξ n = n i= a i ξ i. The appropriate selection of the set of real numbers {a i R, i =,...,n} allows the convenient and in many cases explicit computation of the solution in the transformed coordinates. Therefore, Eq..3 can be replaced by the equation ut = k ξt, where ξt R n is the solution of linear differential equations ξ j = ξ j+ for i =,...,n and ξ n = n j= a j ξ j on the interval t [T i, T i+ with initial condition ξt i = z N. A precise implementation of.3 for a planar nonlinear system is shown in Example 4. below. However, it should be noted that even if the solution map of.3 is not available in explicit form then numerical methods may be used for the computation of the solution of.3. The numerical computation of the solution of.3 is much easier than the computation of the solution of. because we already know that.3 are systems with a globally asymptotically and locally exponentially equilibrium point see [8] for methodologies of exploiting the existence of a globally asymptotically and locally exponentially equilibrium point to produce a qualitatively correct simulation even for stiff differential equations. The structure of the paper is as follows: Sect. 2 provides some results for the numerical explicit Euler scheme for control systems, which are necessary for the proofs of the main results. The results in Sect. 2 are not available in numerical analysis textbooks but their proofs are made in the same way with the corresponding results for systems without inputs. Section 3 is devoted to the proof of Theorem.. Section 4 contains the example of a nonlinear planar system in strict feedback form. Section 5 provides the concluding remarks of the present work. The Appendix contains the proofs of certain auxiliary results.

5 Numerical schemes for nonlinear predictor feedback 523 Notation. Throughout the paper, we adopt the following notation: * For a vector x R n, we denote by x its usual Euclidean norm, by x its transpose. For x R n and ε>0, we denote by B ε x the closed ball or radius ε>0centered at x R n, i.e., B ε x := {y R n : y x ε}. * R + denotes the set of non-negative real numbers. Z + denotes the set of nonnegative integers. For every t 0, [t] denotes the integer part of t 0, i.e., the largest integer being less or equal to t 0. A partition π ={T i } i=0 of R+ is an increasing sequence of times with T 0 = 0 and T i +. * We say that an increasing continuous function γ :R + R + is of class K if γ0 = 0. We say that an increasing continuous function γ :R + R + is of class K if γ0 = 0 and lim s + γs =+.ByKL, we denote the set of all continuous functions σ = σs, t :R + R + R + with the properties: i for each t 0 the mapping σ, t is of class K ; ii for each s 0, the mapping σs, is non-increasing with lim t + σs, t = 0. *ByC j A C j A ;, where A R n R m, j 0 is a non-negative integer, we denote the class of functions taking values in R m that have continuous derivatives of order j on A R n. *Letx:[a r, b R n with b > a 0 and r 0. By T r tx, we denote the open history of x from t r to t, i.e., T r txθ := xt + θ ; θ [ r, 0, for t [a, b. *LetI R + := [0, + be an interval. By L I ; U, we denote the space of measurable and bounded functions u defined on I and taking values in U R m. Notice that we do not identify functions in L I ; U which differ on a measure zero set. For x L [ r, 0;R n, we define x r := sup θ [ r,0 xθ. Notice that sup θ [ r,0] xθ is not the essential supremum but the actual supremum and that is why the quantities sup θ [ r,0] xθ and sup θ [ r,0 xθ do not coincide in general. * The shift operator δ τ u maps each function u : [ τ,0 U to the function δ τ u :[0,τ U with δ τ us = u τ + s for all s [ τ,0. * A function f : A R, where 0 A R n is positive definite if f 0 = 0 and f x >0 for all x = 0. A function f :R n Ris radially unbounded if the set {x R n : f x M} is bounded or empty for every M > 0. 2 Numerical approximation of the solutions of forward complete systems We consider system. under the following assumptions: H f :R n R m R n is a locally Lipschitz vector field with f 0, 0 = 0 that satisfies: f x, u f y, u L x + y + u x y, for all x, y R n, u R m 2. f x, u x + u L x + u, for all x R n, u R m 2.2 where L :R + [, + is a continuous, non-decreasing function. H2 System. is forward complete.

6 524 I. Karafyllis, M. Krstic Assumptions H and H2 have important consequences for system.. Next we point out two consequences which will be used in this section: C There exist a C 2 function W :R n [, + which is radially unbounded, a constant c > 0 and a function p K such that W x f x, u cwx + p u, for all x R n, u R m 2.3 C2 For every τ > 0, there exists a function a τ K such that the solution xt of. with arbitrary initial condition x0 = x 0 0 corresponding to arbitrary measurable and essentially bounded input u :[0,τ R m satisfies where xt a τ x 0 + u, for all t [0,τ] 2.4 u :=ess sup ut t [0,τ Moreover, for every τ>0, there exists a constant M τ > 0 such that a τ s = M τ s for all s [0, ]. The existence of a C 2 function W :R n [, + and the existence of a function a τ K satisfying the requirements of assumption C2 are direct consequences of Theorem, Corollary 2.3 in [] and assumption H. Let P :R + R + be a non-decreasing continuous function that satisfies: { } Ps s 2 L 2 s max 2 W ξ : ξ s + τ Ls, for all s Let Q :R + R + be a non-decreasing continuous function that satisfies: { Qs + max x :W x exp2cτ max y s 2c W y + exp2cτ } ps, for all s Define for all s 0: As := LQs + a τ s + s 2.7 Bs := LQs + a τ s + sa τ s + sla τ s + s 2.8 Consider the following numerical scheme, which is an extension of the explicit Euler method to systems with inputs: we select a positive integer N and define x i+ = x i + i+h ih f x i, us ds, for i = 0,...,N 2.9 for h = τ/n.

7 Numerical schemes for nonlinear predictor feedback 525 Theorem 2. Consider system. under assumptions H, H2. Let τ > 0 be a positive constant and let a C 2 function W :R n [, + which is radially unbounded, a constant c > 0, functions p K, a τ K be such that assumptions C and C2 hold. Let P :R + R +, Q :R + R +, A :R + R +, B : R + R + be continuous functions that satisfy 2.5, 2.6, 2.7, 2.8. Let arbitrary x 0 R n and arbitrary measurable and essentially bounded input u :[0,τ R m. If N τ PQ x 0 + u + u 2c then the following inequalities hold: xτ x N τ B x 0 + u 2NA x 0 + u expτ A x 0 + u 2.0 x i Q x 0 + u, for all i = 0,,...,N 2. where xτ is the solution of. with initial condition x0 = x 0 corresponding to input u :[0,τ R m at time t = τ. Remar.2 Inequality 2.0 shows that if we know the initial condition x0 = x 0 and the applied input u :[0,τ R m then we can estimate all quantities involved in 2.0. Moreover, if we want the approximation error to be less than ε>0itsuffices to select the positive integer N so that: B x0 + u N τ max 2ε A x 0 + u expτ A x 0 + u, PQ x 0 + u + u 2c Notice that the right hand-side of the above inequality can be evaluated before we start applying the scheme 2.9. The restriction is imposed to obtain the uniform bound provided by 2. and it is necessary for the control of the increase of the function W exactly in the same spirit as step size control was applied in [8] for the control of the decrease of the Lyapunov function. The bound provided by 2. is useful for the proof of Theorem.. The proof of Theorem 2. depends on three technical lemmas which are stated below and are proved at the Appendix. Lemma 2.3 Consider system. under the assumptions of Theorem 2.. If x i + u > 0 and h 2cWx i P x i + u, where P :R + R + is the function involved in 2.5, then i+h W x i+ exp2chw x i + exp2cih + h sp us ds 2.2 ih Lemma 2.4 Consider system. under the assumptions of Theorem 2.. If h 2c PQ x 0 + u + u then W x i exp2cihw x 0 + ih 0 exp2cih sp us ds for all where Q :R + R + is the function involved in 2.6. i = 0,..., N 2.3

8 526 I. Karafyllis, M. Krstic Lemma 2.5 Consider system. under the assumptions of Theorem 2.. Define e i := x i xih, i {0,...,N}, where xt is the solution of. with initial condition x0 = x 0 corresponding to input u : [0,τ R m and suppose that 2c h PQ x 0 + u + u. Then e i h2 2 B x 0 + u expiha x 0 + u, for all i {,...,N} 2.4 expha x 0 + u where the functions A, B :R + R + are defined by 2.7, 2.8. We are now ready to provide the proof of Theorem 2.. Proof of Theorem 2. All assumptions of Lemmas 2.4 and 2.5 hold. Consequently, inequalities 2.3, 2.4 hold. Inequality 2.0 follows from using the fact expha x 0 + u ha x 0 + u and definition h = N τ in conjunction with 2.4 fori = N. Moreover, inequality 2.0 implies W x i exp2c τw x 0 + exp2c τ 2c p u. The previous inequality in conjunction with 2.6 implies2.. The proof is complete. Theorem 2. allows us to construct mappings which approximate the solution of. τ time units ahead with guaranteed accuracy level. Indeed, let R C 0 R + ;R + be a positive definite function with lim inf s 0 + Rs s > 0. Define the mapping : R n L [0,τ;R m R n by means of the equation: x 0, u := x N 2.5 where x i, i =,...,N are defined by the numerical scheme 2.9 with h = τ N, N = N x 0 + u and Ns := for s > 0 and [ aτ s+s τ max 2Rs La τ s+sexpτ As, PQs+s ] + 2c 2.6 N0 := 2.7 By virtue of 2.0, the mapping :R n L [0,τ;R m R n satisfies x 0, u xτ R x 0 + u 2.8 Inequalities 2.0, 2. in conjunction with 2.8 and 2.4 imply the following inequality: x 0, u minr x 0 + u + a τ x 0 + u, Q x 0 + u 2.9 Notice that the mapping Ns defined by 2.6 and 2.7 is locally bounded. Indeed, there exists a constant M τ > 0 such that a τ s = M τ s for all s 0 sufficiently small.

9 Numerical schemes for nonlinear predictor feedback 527 Therefore, continuity of all functions involved in 2.6 in conjunction with the fact that lim inf s 0 +Rs/S >0 implies that Therefore, we conclude: sup Nl <+, for all s l s Corollary 2.6 Consider system. under the assumptions of Theorem 2.. For every positive definite function R C 0 R + ;R + with lim inf s 0 +Rs/S >0 and for every τ > 0, consider the mapping : R n L [0,τ;R m R n defined by 2.5 for all x 0, u R n L [0,τ;R m, where x i, i =,...,N are defined by the numerical scheme 2.9 with h = τ N and N := N x 0 + u, where N :R + {, 2, 3,...} is defined by 2.6, 2.7. Then inequalities 2.8, 2.9 hold for all x 0, u R n L [0,τ;R m, where xt denotes the solution of. with initial condition x0 = x 0 corresponding to input u :[0,τ R m and u :=ess sup t [0,τ ut Moreover, inequality 2.20 holds for all s 0. 3 Proof of Theorem. This section is devoted to the proof of Theorem.. The proof of Theorem. is constructive and formulae will be given next for the locally bounded mapping N : R + {, 2, 3,...} involved in the hybrid dynamic feedback law defined by.3,.4 and.5. To simplify the procedure of the proof, we break the proof up into two steps. First step: Construction of feedback Second step: Rest of proof The control practitioner, who is not interested in reading the details of the proof, may read only the first step of the proof. 3. First step: Construction of feedback The feedback law is entirely given by.3.5, except for the function N :R + {, 2, 3,...}, whose construction is given here. We assume the knowledge of a function L :R + [, +, ac 2 function W :R n [, + and a function a τ K satisfying the requirements of assumptions C, C2 of Sect. 2. As remarked in the previous section, the existence of a function L :R + [, +, ac 2 function W : R n [, + and a function a τ K satisfying the requirements of assumptions H, C, C2 are direct consequences of Theorem, Corollary 2.3 in [] and the fact that f :R n R m R n is a continuously differentiable mapping with f 0, 0 = 0. Moreover, we need to assume the knowledge of a Lyapunov function for the closedloop system. with ut = kxt. More specifically, we assume the existence of a positive definite, radially unbounded function V C R n ;R + for 2., constants ε, K,μ>0 and a function ρ K such that the following hold:

10 528 I. Karafyllis, M. Krstic V x f x, kx ρv x, x R n 3. x 2 V x K x 2, x B ε V x 2K x, x B ε V x f x, kx μ x 2, x B ε The existence of a Lyapunov function for the closed-loop system. with ut = kxt satisfying 3., 3.2, 3.3, 3.4 is a direct consequence of Proposition 4.4 in [8]. Based on the knowledge of all the functions and constants described above, we next proceed to the construction of new functions. The first functions to define are the continuous, non-decreasing functions P :R + R +, Q :R + R +, A :R + R +, B :R + R + that satisfy 2.5, 2.6, 2.7, 2.8. Next, we define: functions a i K i =,...,4 and constants k,, k 3, k 4 > 0 that satisfy: a x V x a 2 x, for all x R n 3.5 V x a 3 x and kx a 4 x, for all x R n 3.6 a s := k s 2, a 2 s := s 2, a 3 s := k 3 s, a 4 s := k 4 s, for all s [0,ε] 3.7 a continuous, non-decreasing function M :R + [, + that satisfies: f x, kz f x, kx M x + z z x, for all z, x R n 3.8 The reader should notice that the existence of functions M :R + [, +, a i K i =,...,4 and constants k,, k 3, k 4 > 0 satisfying 3.5, 3.6, 3.7 and 3.8 is a direct consequence of a the fact that V C R n ;R + is positive definite and radially unbounded see Lemma 3.5 in [], b of Lemma 2.4 in [7], c of inequalities 3.2, 3.3 and d of the fact that f :R n R m R n and k :R n R m are continuously differentiable mappings with k0 = 0. Moreover, define for all s 0: D r s := a 3 a r s + s Ma r s + s exprla r s + s, qs := a 4 a a 2s + a a 2s 3.9 where a r K is the function involved in 2.4 with τ replaced by r > 0 and L :R + [, + is the function involved in assumption H. Next select a constant δ>0, such that: a a 2 2a δ ε 3.0

11 Numerical schemes for nonlinear predictor feedback 529 Having selected δ>0, we are in a position to select a constant γ>0, so that: γ min a δ, δ 2 ρ 2 3. Define: φ := k 3 M L := L a a 2 + k 4 a 2a δ a 2 2a δ + a δ + a δ expr L and moreover, select a constant R > 0, so that: Finally, define: k 4 k R <, 2k2 φ k + k 4 k2 + μk k 4 k2 μk k R < 3.4 γ Rs := min max, D r a τ s + qqs, Rs, 2 a 2 a a 4 s Notice that by virtue of 3.7 and the fact that Qs for all s 0, it follows from definition 3.5 that lim inf s 0 + Rs s = min R, 4k k 4 >0. Therefore, Corollary 2.6 guarantees that the mapping N :R + {, 2, 3,...} defined by 2.6, 2.7 is locally bounded and the mapping :R n L [0,τ;R m R n defined by 2.5 satisfies inequalities 2.8, 2.9 for all x 0, u R n L [0,τ;R m, where xt denotes the solution of. with initial condition x0 = x 0 corresponding to input u :[0,τ R m and u :=ess sup t [0,τ ut. 3.2 Second step: rest of proof Having completed the design of the feedback law by constructing the function N : R + {, 2, 3,...} in 2.6, we are now ready to prove some basic results concerning the closed-loop system.2 with.3,.4,.5 and 2.6. The following claim shows that practical stabilization is achieved. Its proof is provided in the Appendix. Claim There exists σ K L such that for every partition {T i } i=0 of R + with sup i 0 T i+ T i r, for every x 0 R n and u 0 L [ τ,0;r m, the solution of.2,.3,.4 and.5 with initial condition x0 = x 0, T τ 0u = u 0 satisfies the following inequality for all t 0: V xt max σ x 0 + u 0 τ, t, ρ 2γ 3.6

12 530 I. Karafyllis, M. Krstic where ρ K is the function involved in 3. and γ>0 is the constant involved in 3. and 3.5. The following claim shows that local exponential stabilization is achieved. Its proof is provided in the Appendix. Claim 2 There exist constants Q, Q 2,ω >0 such that for each partition {T i } i=0 of R + with sup i 0 T i+ T i r, for each x 0 R n and u 0 L [ τ,0;r m,the solution of.2,.3,.4 and.5 with initial condition x0 = x 0, T τ 0u = u 0 satisfies the following inequalities: ut expω t T j Q sup xw + T τ T j u τ, for all t T j 3.7 T j w T j +τ xt expω t T j τ Q 2 sup xw + T τ T j u, for all t T j + τ T j w T j +τ τ 3.8 where T j is the smallest sampling time for which it holds V xt j + τ δ, where δ>0 is the constant involved in 3.0 and 3.. The following claim guarantees that u is bounded. Its proof is provided in the Appendix. Claim 3 There exists a non-decreasing function G :R + R + such that for each partition {T i } i=0 of R + with sup i 0 T i+ T i r, for each x 0 R n and u 0 L [ τ,0;r m, the solution of.2,.3,.4 and.5 with initial condition x0 = x 0, T τ 0u = u 0 satisfies the following inequality for all t 0: xt + T τ tu G x 0 + u 0 τ 3.9 τ We are now ready to prove Theorem.. Let arbitrary partition {T i } i=0 of R + with sup i 0 T i+ T i r, x 0 R n, u 0 L [ τ,0;r m and consider the solution of.2,.3,.4 and.5 with arbitrary initial condition x0 = x 0, T τ 0u = u 0. Inequalities 3.5 and 2.4 imply that the smallest sampling time T j for which V xt j + τ δ holds is T 0 = 0 for the case a 2 a τ x 0 + u 0 τ δ. Moreover, the fact that there exists a constant M τ > 0 such that a τ s = M τ s for all s [0, ], in conjunction with inequalities 3.7, 3.8, allow us to conclude that there exists a constant >0 xt + T τ tu exp ω t x 0 + u 0 τ, for all t τ M τ a provided that x 0 + u 0 τ min, 2 δ. Proposition 7 in [6] in conjunction with 3.6, 3. and the fact that sup i 0 T i+ T i r allow us to guarantee the existence of a non-decreasing function T :R + R + such that the smallest sampling time T j for which V xt j +τ

13 Numerical schemes for nonlinear predictor feedback 53 δ holds satisfies T j T x 0 + u 0 τ for all x 0, u R n L [0,τ;R m.combining 3.7, 3.8, 3.9 with the previous inequality, allows us to conclude the existence of a non-decreasing function G :R + R + such that the following inequality holds for all t 0: xt + T τ tu exp ω t G x 0 + u 0 τ 3.2 τ Consequently, using 3.20 and 3.2, we conclude that.6 holds with Cs := Cw dw for all s > 0 and C0 := 0, where s 2s s Cs := max, Gl s, for all s [0, l] l Cs := max s, Gs, for all s > l l := min, a2 M δ τ The proof of Theorem. is complete. 4 An illustrative example This section is devoted to the presentation of an example of a planar nonlinear system in strict feedback form. As noted in the introduction the analog component of the proposed control scheme can be implemented with precision by utilizing transformed coordinates. Example 4. Consider the two-dimensional control system ẋ t = a + cosx tx t + x 2 t ẋ 2 t = ut τ 4. where a > 0 is a constant. System 4. is a nonlinear system which is not globally Lipschitz and consequently, the predictor feedback proposed in [0] cannot be used. Moreover, the solution map for system 4. is not available and consequently, the predictor feedback proposed in [9] cannot be used. However, we show next that Theorem. can be applied for system 4.. Moreover, following the first step of Theorem., we give next explicit formulas for the feedback law. [ We first notice that ] inequalities 2., 2.2 hold for the vector field f x, u := a+cosx x +x 2 with Ls := +2a +as. Property C holds with W x := u + 2 x2 + 2 x2 2. More specifically, using the inequalities x x 2 2 x2 + 2 x2 2, x 2u 2 x u2, we obtain inequality 2.3 with c := 22a + and ps := 2 s2. Using the inequalities x x 2 2 x2 + 2 x2 2, x 2u 2 x u2 for the function W x := 2 x2 + 2 x2 2, we obtain the differential inequality W x f x, u

14 532 I. Karafyllis, M. Krstic 22a + W x + 2 u2 for all x, u R 2 R. Utilizing the previous differential inequality along the solutions of 4. with τ = 0, we obtain the following estimate: W xt exp2 + 2at W x0 + 2 t 0 exp22a + t su 2 s ds 4.2 for all t 0 for which the solution of 4. with τ = 0 exists. Estimate 4.2 allows us by means of a standard contradiction argument to conclude that the solution of 4. with τ = 0 exists for all t 0 and for all initial conditions and applied inputs. Moreover, inequality 4.2 implies the following estimate: exp2a + t xt exp2a + t x0 + sup us, for all t a + 0 s t It follows that property C2 holds with a τ s := s exp2a + τ and M τ := exp2a + τ for all τ 0. We next define: V x := 4a + 2 x 2 +x 2 + a + x +ax cosx 2, for all x R kx := a a cosx ax sinx x 2 + ax + ax cosx x, for all x R Using the inequality a + + a cosx x x 2 2 ε a + + a cosx 2 2ε x2 2 with ε = a++a cosx + a++a cosx a++a cosx in conjunction with definition 4.4, we obtain for all x R 2 : V x = 4a+ 2 +a + +a cosx 2 x 2 + x2 2 +2a + + a cosx x x 2 4a εa + + a cosx 2 x 2 + x2 2 ε a + + a 4a + 2 cosx a + + a cosx a + + a cosx + x a + + a cosx x2 2 4a a a + x 2a x2 2 4a + 2 2a + 2 x 2 + 2a x2 2 x Moreover, using the inequality cosx, the triangle inequality and completing the squares, we obtain directly from Definition 4.4 for all x R 2 : V x 8a a + 2 x 2 + x2 2 64a + 4 x 2 4.7

15 Numerical schemes for nonlinear predictor feedback 533 and V x 8a + 2 x +8a + 2 x 2 + a + x + ax cosx a + + a cosx ax sinx +8a + 2 x 2 + a + x + ax cosx 8a + 3 4a a x x 4.8 Finally, using the inequality x x 2 + a + x + ax cosx 2 x2 + 2 x 2 +a+ x + ax cosx 2, we obtain directly from definitions 4.4, 4.5 for all x R 2 : V x f x, kx V x 4.9 It follows from inequalities 4.6, 4.7, 4.8, 4.9 that inequalities 3., 3.2, 3.3, 3.4 hold with ρs := s,μ :=, K := 64a + 4 and ε :=. Moreover, inequalities 4.6, 4.7, 4.8, 4.9 and definition 4.5 imply that inequalities 3.5, 3.6, 3.7, 3.8 hold with a s := s 2, k :=, a 2 s := 64a + 4 s 2, := 64a + 4, a 3 s := 48a + 3 s max, s, k 3 := 48a + 4, a 4 s := 6a + 2 s max, s and k 4 := 6a + 2. Definition 4.5 implies that inequality 3.8 holds with Ms := 8a 2 + 6a + 3s 2 + s + since the following inequality holds for all x R 2 : kx 8a 2 + 6a + 3 x 2 + x All previous definitions allow us to conclude that 3.0, 3., 3.2, 3.3 and 3.4 hold with δ :=,γ :=, L := + 3a + 256a a+ 4 a 6a + 2 +,φ:= 48a + 4 8a 2 + 6a+3 6a a+2 + 6a+ 2 6a+ 2 6a+ 2 expr L and R =. Finally, define for all s 0: 32a a+ 4 φ+3 s := 48a + 3 s max, s Ms exprls, qs := 8a + 2 s 6a + 2 max, 8a + 2 s + 4. Rs := 32a + 4 min 32 max, D r a τ s + qqs, s a + 4 φ + 3, 6 mins, 2a + 3s 4.2 Ps := s 2 + 2a + as 2, Qs:=2exp2+2aτs+, D r s:= a r s+s 4.3

16 534 I. Karafyllis, M. Krstic As := + 2a + 2a exp22a + τ + as + exp2a + τ +2exp22a + τ 4.4 Bs := LQs + a τ s + sa τ s + sla τ s + s 4.5 [ ] aτ s + s PQs + s Ns := τ max La τ s + sexpτ As, 2Rs 42a + + for, s > 0 N0 := 4.6 where Ls := + 2a + as, Ms := 8a 2 + 6a + 3s 2 + s +, a r s := s exp2a + r and a τ s := s exp2a + τ. It follows from the proof of Theorem. that for every τ > 0, r > 0 there exists a constant ω>0 and a locally Lipschitz, non-decreasing function C :R + R + with C0 = 0, such that for every partition {T i } i=0 of R + with sup i 0 T i+ T i r, for every x 0 R 2 and u 0 L [ τ,0;r, the solution xt, ut R 2 Rof the closed-loop system 4. with ż t = a + cosz tz t + z 2 t fort [T i, T i+ ż 2 t = ut 4.7 ut = a a cosz t az t sinz tz 2 t + az t +az t cosz t z t 4.8 and where N := N xt i +sup Ti τ s<t i us, h = τ N and zt i = z N 4.9 ah + cosz j, z j, + hz j,2 j+h z j+ = z j + ut i τ + s ds, for j = 0,...,N z 0 = xt i jh 4.20 and initial condition x0 = x 0 and us = u 0 s for s [ τ,0 satisfies inequality.6 for all t 0. The analog component 4.7 of the hybrid predictor feedback law 4.7, 4.8, 4.9, 4.20 can be implemented by utilizing the fact that system 4.7, 4.8 expressed in the coordinates satisfies the following differential equations ξ = z ξ 2 = a + z + a cosz z + z 2 4.2

17 Numerical schemes for nonlinear predictor feedback 535 ξ t = ξ t + ξ 2 t ξ 2 t = ξ 2 t 4.22 It follows from 4.2, 4.22 that the analog component 4.7 of the hybrid predictor feedback law 4.7, 4.8, 4.9, 4.20 can be implemented by means of the equations: ξ t = exp t T i z T i + t T i a + z T i + a cosz T i z T i + z 2 T i, for t [T i, T i ξ 2 t = exp t T i a + z T i + a cosz T i z T i + z 2 T i z t = ξ t z 2 t = ξ 2 t a + ξ t a cosξ tξ t 4.24 The methodology for handling feedback linearizable systems in the strict feedback form.7 is similar to the methodology described above for system Concluding remarks This work has focused on a key aspect of the implementation problem for predictorbased feedback laws: the problem of the approximation of the predictor mapping. It was shown that the numerical approximation of the predictor mapping by means of the explicit Euler numerical scheme in conjunction with a hybrid feedback law that uses sampled measurements can be used for the global stabilization of all forward complete nonlinear systems that are globally asymptotically stabilizable and locally exponentially stabilizable in the delay-free case. The present paper goes beyond the approximation results in [6] by removing the global Lipschitz restriction. More remains to be done for the approximations of the integrals involved in the explicit Euler scheme by easily implementable formulae. Furthermore, one cannot ignore the possibility of using different numerical schemes except the explicit Euler scheme; see [3]: the use of implicit numerical schemes may require fewer grid points than the grid points needed for the explicit Euler scheme. Finally, there is the challenging problem of using numerical approximations for cases where the measured output is not necessarily the state vector and there is a measurement delay see [9,0]. Appendix Proof of Lemma 2.3: Define the function: gλ = W x i + λx i+ x i 6.

18 536 I. Karafyllis, M. Krstic for λ [0, ]. The following equalities hold for all λ [0, ]: dg dλ λ = W x i + λx i+ x i x i+ x i d 2 g dλ 2 λ = x i+ x i 2 W x i + λx i+ x i x i+ x i 6.2 Moreover, notice that by virtue of 2.2 and 2.9, it holds that x i+ x i h x i + u L x i + u. The previous inequality in conjunction with 2.5 and 6.2gives: d 2 g dλ 2 λ h2 P x i + u 6.3 where P :R + R + is the function involved in 2.5. Furthermore, inequality 2.3 in conjunction with 2.9 and 6.2 gives: i+h dg dλ 0 = W x i f x i, us ds chwx i + ih i+h ih p us ds 6.4 Combining 6., 6.3 and 6.4, we get: W x i+ = g + chw x i + i+h ih p us ds + h2 2 P x i + u 6.5 Inequality 6.5 in conjunction with the following inequality + chw x i + + i+h ih i+h ih p us ds + h2 2 P x i + u exp2chw x i exp2cih + h sp us ds which holds for all h 2cWx i P x i + u imply that 2.2 holds. The proof is complete. Proof of Lemma 2.4 We will first prove that if there exists j {0,...,N } such 2c that u +min i=0,..., j x i > 0 and h PQ x 0 + u + u then 2.3 holds for all i = 0,..., j +. The proof is by induction. First notice that 2.3 holds for i = 0. Suppose that it holds for some i {0,..., j}. exp2c τ Clearly, inequality 2.3 implies W x i exp2c τw x 0 + 2c p u. The previous inequality in conjunction with 2.6 implies x i Q x 0 + u. Consequently, the facts that P :R + R + is non-decreasing and W x i imply

19 Numerical schemes for nonlinear predictor feedback 537 2c h PQ x 0 + u + u 2cWx i P x i + u. Since x i + u > 0 and h 2cWx i P x i + u, Lemma 2.3 shows that: W x i+ exp2chw x i + i+h ih exp2cih + h sp us ds The above inequality in conjunction with 2.3 shows that 2.3 holds for i replaced by i +. The case that there exists j {0,...,N } with u +min i=0,..., j x i =0can be treated in the following way. Let j {0,...,N } be the smallest integer with u +min i=0,..., j x i =0. This implies that u =0 and x j =0. Since f 0, 0 = 0, 2.9 impliesthat x i =0for all i = j +,...,N. Consequently, 2.3 holds for all i = j +,...,N. The proof is complete. Proof of Lemma 2.5 Notice that, by virtue of 2.9, the following equation holds for all i {0,...,N }: e i+ = e i + i+h ih f x i, us f xs, us ds 6.6 Inequality 2. implies the following inequality for all i {0,...,N } and s [ih,i + h]: f x i, us f xs, us L x i + xs + u x i xs 6.7 Using the definition e i := x i xih and inequalities 2.2, 2.4, we get for all i {0,...,N } and s [ih,i + h]: x i xs e i + xs xih e i +s ih max xl + u L max xl + u ih l s ih l s e i +s ih a τ x 0 + u + u L a τ x 0 + u + u 6.8 Notice that all hypotheses of Lemma 2.4 hold. Therefore, inequality 2.3 holds for all i = 0,...,N. Clearly, inequality 2.3 implies W x i exp2c τw x 0 + exp2c τ 2c p u. The previous inequality in conjunction with 2.6 implies x i Q x 0 + u for all i = 0,...,N. Exploiting the fact that x i Q x 0 + u for all i = 0,...,N and 6.6, 6.7, 6.8, we obtain for all i {0,...,N }: e i+ e i +hlq x 0 + u + a τ x 0 + u + u e i + h2 2 LQ x 0 + u + a τ x 0 + u + u a τ x 0 + u + u La τ x 0 + u + u 6.9

20 538 I. Karafyllis, M. Krstic Definitions 2.7, 2.8 in conjunction with inequality 6.9 shows that the following recursive relation holds for all i {0,...,N } e i+ expha x 0 + u e i + h2 2 B x 0 + u 6.0 Using the fact e 0 = 0, in conjunction with relation 6.0, gives the desired inequality 2.4. The proof is complete. Proof of claim First, we show that for each partition {T i } i=0 of R + with sup i 0 T i+ T i r, for each x 0 R n and u 0 L [ τ,0;r m,thesolution of.2,.3,.4 and.5 with initial condition x0 = x 0, T τ 0u = u 0 is unique and exists for all t 0. The solution of.2,.3,.4 and.5 is determined by the following process: Initial step: Given x0 = x 0, T τ 0u = u 0 we determine the solution xt of.2 for t [0,τ]. Notice that the solution is unique. Inequality 2.4 implies the following estimate: xt a τ x 0 + u 0 τ, for all t [0,τ] 6. i-th step: Given xt for t [0, T i + τ] and ut for t [ τ,t i, we determine xt for t [0, T i+ + τ] and ut for t [ τ,t i+. The solution zt of.3 for t [T i, T i+ with initial condition zt i = z N is unique by virtue of the fact that f and k are locally Lipschitz mappings. Inequality 3. implies: V zt V zt i, for all t [T i, T i+ 6.2 We determine ut for t [T i, T i+ by means of the equation ut = kzt. Notice that inequalities 3.5, 3.6 in conjunction with 6.2 imply the following inequality for all t [T i, T i+ : ut = kzt a 4 a a 2 zt i 6.3 Finally, we determine the solution xt of.2 fort [T i + τ,t i+ + τ]. Notice that the solution is unique. The fact that T i+ T i r in conjunction with inequality 2.4 with τ replaced by r > 0 and inequality 6.3 implies the estimate: xt a r xt i + τ +a 4 a a 2 zt i, for all t [T i + τ,t i+ + τ] 6.4

21 Numerical schemes for nonlinear predictor feedback 539 Next we evaluate the difference zt xt + τfor t [T i, T i+. Exploiting 2. we get: t zt xt + τ = zt i xt i + τ+ f zs, kzs f xs + τ,kzs ds t zt i xt i + τ + T i T i L zs + xs + τ + kzs zs xs + τ ds Using the right inequality 3.5, inequalities 6.2, 6.3, 6.4, in conjunction with the above inequality, we obtain: zt xt + τ zt i xt i + τ +La a 2 zt i +a 4 a a 2 zt i + a r xt i + τ +a 4 a a 2 zt i t zs xs + τ ds T i Define ϕs := a r s + s. Using the Growall Bellman lemma, the above inequality and the fact that T i+ T i r, we get for all t [T i, T i+ : zt xt + τ zt i xt i + τ exprlϕ xt i + τ +q zt i 6.5 Next we evaluate the quantity V xt +τf xt +τ,kzt for t [T i, T i+. Using inequality 3. we get: V xt + τf xt + τ,kzt ρv xt + τ + V xt + τ f xt + τ,kzt f xt + τ,kxt + τ The following estimate follows from 3.6, 3.8 and the above inequality: V xt + τf xt + τ,kzt ρv xt + τ +a 3 xt + τ M t + τ + zt xt + τ zt Using the above inequality in conjunction with inequality 3.5, inequalities 6.2, 6.4 and definitions qs := a 4 a a 2s+a a 2s, ϕs := a r s+s, we get: V xt + τf xt + τ,kzt ρv xt + τ + a 3 ϕ xt i +τ +q zt i Mϕ xt i +τ +q zt i xt + τ zt 6.6

22 540 I. Karafyllis, M. Krstic Combining inequalities 6.5, 6.6 and definition 3.9, we obtain for all t [T i, T i+ : V xt + τf xt + τ,kzt ρv xt + τ + D r xt i + τ +q zt i xt i + τ zt i 6.7 Since zt i = z N recall 2.4, it follows from 2.8, 2.9 and.2,.4 that the following inequalities hold for all i = 0,, 2,...: zt i xt i + τ R xt i + T τ T i u τ zt i Q xt i + T τ T i u τ Since xt i + τ a τ xt i + T τ T i u τ recall 2.4, we obtain from 6.7, 6.8, 6.9 and definition 3.5 for all t [T i, T i+ : d V xt + τ ρv xt + τ + γ 6.20 dt Using 6.20 and Lemma 2.4, page 82 in [7], we obtain for all t 0: V xt + τ max σv xτ, t, ρ 2γ 6.2 for certain function σ KL. Combining 3.5, 6. and 6.2, we obtain inequality 3.6 with σs, t := σa 2 a τ s, t τ for all t > τ and σs, t := σa 2 a τ s, 0 for all t [0,τ]. The proof is complete. Proof of Claim 2 Let arbitrary partition {T i } i=0 of R + with sup i 0 T i+ T i r, x 0 R n, u 0 L [ τ,0;r m and consider the solution of.2,.3,.4 and.5 with arbitrary initial condition x0 = x 0, T τ 0u = u 0. Inequalities 3. and 3.6 guarantee that there exists a unique smallest sampling time T j such that V xt j + τ δ. Moreover, inequalities 6.20, 3. and 3.5 allow us to conclude that xt a δ and V xt δ, for all t T j + τ 6.22 Using 6.8, definition 3.5, 3. and 6.22, we obtain for all i j: zt i zt i xt i + τ + xt i + τ γ + a δ 2a δ 6.23

23 Numerical schemes for nonlinear predictor feedback 54 Using 6.2, 3.5 and 6.23, we get for all t T j : zt a a 22a δ 6.24 Next we evaluate the difference zt xt + τ for t T j. Exploiting 2. and inequalities 3.6, 3.7, 3.0, 6.22, 6.24 and definition 3.3, we get for all i j and t [T i, T i+ : zt xt + τ = zt i xt i + τ+ t t zt i xt i + τ + L T i f zs, kzs f xs + τ,kzs ds T i zs xs + τ ds Using the Growall Bellman lemma, the above inequality and the fact that T i+ T i r imply for all i j and t [T i, T i+ : zt xt + τ zt i xt i + τ exp r L 6.25 Next we evaluate the quantity V xt +τf xt +τ,kzt for t [T i, T i+. Using inequalities 3.4, 3.5, 3.7, 6.22, 3.0, 3.8, 6.24 and 6.25 and definition 3.2, we get for all i j and t [T i, T i+ : V xt + τf xt + τ,kzt μk2 V xt + τ +φ xt + τ xt i + τ zt i 6.26 Using 3.5, 3.7, 3.0, 6.22 and 6.26, we get for all i j and t [T i, T i+ : V t + τ μ 2 V t + τ+ 2μk φ 2 xt i + τ zt i where V t = V xt. Using6.8, 3.5 and 6.27, we get for all i j and t [T i, T i+ : V t + τ μ 2 V t + τ+ μk φ 2 R 2 xt i 2 + μk φ 2 R 2 T τ T i u 2 τ 6.28 Let ω< μ 4 be a positive constant sufficiently small so that k 4 R expω r + τ < and 2 expωr + τ φ R k k μ 2 4ωμ + k 4 k2 expωr + τ R + exp ωτ < 6.29 k Rk 4 k2 expωr + τ

24 542 I. Karafyllis, M. Krstic The existence of 0 <ω< μ 4 satisfying 6.29 is guaranteed by 3.4. Using 6.28 and the fact that sup i 0 T i+ T i r, we obtain for all i j and t [T i, T i+ : V t + τ μ V t + τ+ φ 2 R 2 exp 2ωt exp2ωr sup exp2ωs xs 2 2 μk T i s t + μk φ 2 R 2 exp 2ωt exp2ωr + τ sup exp2ωs us T i τ s t The differential inequality 6.30 allows us to conclude that the following differential inequality holds for t T j a.e.: V t + τ μ V t + τ+ φ 2 R 2 exp 2ωt exp2ωr sup exp2ωs xs 2 2 μk T j s t + μk φ 2 R 2 exp 2ωt exp2ωr + τ sup exp2ωs us T j τ s t Integrating 6.3 and since ω< μ 4, we obtain for all t T j : V t +τ exp 2ωt T j V T j + τ+ 2k2 2 μk φ 2 R 2 exp 2ωt μ 4ω exp2ωr sup T j s t exp2ωs xs 2 + 2k2 2 φ 2 R 2 exp 2ωt exp2ωr + τ μk μ 4ω sup T j τ s t exp2ωs us Using 3.5, 3.7, 6.2 and the fact that ω< μ 4, we obtain from 6.32 for all t T j : xt + τ expωt + τ expωt j + τ xt j + τ k + 2 expωr + τ φ R sup expωs xs k μ 2 4ωμ T j s t + 2 expωr + 2τ φ R sup expωs us 6.33 k μ 2 4ωμ T j τ s t Using 3.5, 3.6, 3.7, 3.0, 3.5, 6.2, 6.8 and 6.24 we obtain for all i j and t [T i, T i+ :

25 Numerical schemes for nonlinear predictor feedback 543 ut = kzt k 4 zt k 4 k zt i k 4 k zt i xt i + τ +k 4 Rk 4 k xt i + Rk 4 k xt i + τ k T τ T i u τ + k 4 k xt i + τ 6.34 Inequality 6.34 in conjunction with the fact that sup i 0 T i+ T i r implies: ut expωt Rk 4 expωr xt i expωt i k + Rk 4 expωr + τ sup expωs us k T i τ s<t i +k 4 k expωr τ xt i + τ expωt i + τ Therefore, we get from the above inequality for all t T j : ut expωt k 4 expωr R + exp ωτ sup expωs + τ xs + τ k T j τ s t + Rk 4 expωr + τ sup expωs us 6.35 k T j τ s t Distinguishing the cases sup Tj τ s t expω s us =sup T j s t expω s us and sup Tj τ s t expω s us = sup T j τ s<t j expω s us, we obtain from 6.35 for all t T j : ut expωt k 4 expωr R + exp ωτ k Rk 4 k2 expωr + τ + Rk 4 expωr + τ k sup expωs + τ xs + τ T j τ s t sup expωs us 6.36 T j τ s<t j Combining 6.33 and 6.36, we get for all t T j : xt + τ expωt + τ expωt j + τ xt j + τ k + 2 expωr + τ φ R + k 4 k2 expωr + τ R + exp ωτ k μ 2 4ωμ k Rk 4 k2 expωr + τ

26 544 I. Karafyllis, M. Krstic sup expωs + τ xs + τ T j τ s t + 2 expωr + 2τ φ R k μ 2 4ωμ sup expωs us T j τ s<t j Distinguishing the cases sup Tj τ s t expωs + τ xs + τ = sup T j τ s T j expωs + τ xs + τ, sup Tj τ s t expωs + τ xs + τ = sup T j s t expωs + τ xs + τ and using the above inequality, we obtain for all t T j : xt + τ expωt + τ expωt j + τ xt j + τ A k +A sup Tj τ s T j expωs + τ xs + τ + 2 k A φ R expωr + 2τ μ 2 4ωμ sup T j τ s<t j expωs us Inequalities where A = 2 k φ R expωr+τ + k 4 k2 expωr+τ R+exp ωτ μ 2 4ωμ k Rk 4 k2 expωr+τ 6.36, 6.37 imply that there exist constants Q, Q 2 > 0 such that 3.7, 3.8 hold. The proof is complete. Proof of Claim 3 Let arbitrary partition {T i } i=0 of R + with sup i 0 T i+ T i r, x 0 R n, u 0 L [ τ,0;r m and consider the solution of.2,.3,.4 and.5 with arbitrary initial condition x0 = x 0, T τ 0u = u 0. Define: bs := a 4 a a 2s, for all s and notice that b K. Moreover, notice that definitions 6.38 and 3.5 imply that Rs s 2 b, for all s Furthermore, definition 6.38 and inequality 6.3 imply the following inequality for all i Z + and t [T i, T i+ : ut b zt i 6.40 Inequalities 3.5, 3.6 imply the existence of a non-decreasing function g : R + R + such that: xt g x 0 + u 0 τ, for all t 0 6.4

27 Numerical schemes for nonlinear predictor feedback 545 By virtue of 6.8, 6.39 and 6.4 wegetforalli Z + : zt i xt i + τ R xt i + sup us T i τ s<t i 2 b 2 xt i + sup us 2 T i τ s<t i max 2 b xt i, 2 b sup us T i τ s<t i max 2 b g x 0 + u 0 τ, 2 b sup us T i τ s<t i The above inequality in conjunction with 6.4 givesforalli Z + : zt i xt i + τ +max 2 b g x 0 + u 0 τ, 2 b sup us T i τ s<t i g x 0 + u 0 τ + max 2 b g x 0 + u 0 τ, 2 b sup us T i τ s<t i max2g x 0 + u 0 τ, b g x 0 + u 0 τ, b sup us T i τ s<t i max2g x 0 + u 0 τ, b g x 0 + u 0 τ, b sup us τ s<t i Furthermore, using 6.40 and the above inequality, we obtain for all i Z + : sup T i s<t i+ us max g x 0 + u 0 τ, sup us 6.42 τ s<t i where gs := maxgs, b2gs for all s 0, is a non-decreasing function. Define the sequence: F i := sup us 6.43 τ s<t i Notice that definition 6.43 and the fact that sup τ s<ti+ us = max sup Ti s<t i+ us, sup τ s<ti us in conjunction with 6.42 imply the following inequality for all i Z + : F i+ max g x 0 + u 0 τ, F i 6.44

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