H Control of Linear Singularly Perturbed Systems with Small State Delay

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1 Journal of Mathematical Analysis and Applications 25, 4985 Ž 2. doi:1.16jmaa , available online at on Control of Linear Singularly Perturbed Systems with Small State Delay Valery Y. Glizer Faculty of Aerospace Engineering, TechnionIsrael Institute of Technology, aifa 32, Israel and Emilia Fridman Department of Electrical EngineeringSystems, Tel Ai Uniersity, Ramat Ai 69978, Tel Ai, Israel Submitted by George Leitmann Received April 1, 2 An infinite horizon state-feedback control problem for singularly perturbed linear systems with a small state delay is considered. An asymptotic solution of the hybrid system of Riccati-type algebraic, ordinary differential, and partial differential equations with deviating arguments, associated with this problem, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original problem, independent of the singular perturbation parameter, are derived. A simplified controller with parameter-independent gain matrices, solving the original problem for all sufficiently small values of this parameter, is obtained. An illustrative example is presented. 2 Academic Press 1. INTRODUCTION For many years, controlled systems with disturbances Ž uncertainties. in dynamics have been extensively studied Žsee e.g. 2 and the list of references therein.. One of the main problems in this topic which has been solved is constructing a feedback controller independent of the disturbance, which provides a required property of the closed-loop system for all realizations of the disturbance from a given set. Two classes of distur X $35. Copyright 2 by Academic Press All rights of reproduction in any form reserved.

2 5 GLIZER AND FRIDMAN bances are usually distinguished: Ž. 1 disturbances belonging to a known bounded set of Euclidean space; and Ž. 2 quadratically integrable disturbances. In this paper, we deal with the second class of disturbances. For controlled systems with quadratically integrable disturbance, the problem is frequently considered Žsee e.g. 1, 4.. The control problem has been considered for systems without and with delay in the state variables Žsee e.g. 1, 3, 4, 9, 15.. For both types of systems, the solution of the control problem can be reduced to a solution of a game-theoretic Riccati equation. In the case of undelayed systems, the Riccati equation is finite dimensional, while in the case of delayed systems it is infinite dimensional. The infinite dimensional Riccati equation can be reduced to a hybrid system of three matrix equations of Riccati type. Solving this system is a very complicated problem. In various fields of science and engineering, systems with two-time-scale dynamics are often investigated. Mathematically, such systems are modelled by singularly perturbed equations Žsee e.g. 13, 29.. Control problems for singularly perturbed equations have been extensively investigated for many years Žsee 2, 18, 19, 21, 26, 28 and the references therein.. owever, most of these Ž and more recent. publications are devoted to problems with undelayed dynamics. Singularly perturbed control problems for systems with delays are less investigated. As far as is known to the authors, there are only few publications in this area 7, 8, 112, 24, 25. In the present paper, we consider an infinite horizon state-feedback control problem for singularly perturbed linear systems with a small state delay. The control problem for singularly perturbed systems without delays has been studied in a number of papers 5, 17, 22, 23, 27, 3. owever, as far as is known to the authors, the control problem for singularly perturbed systems with delays has not yet been considered in the open literature. The main results, obtained in this paper, are: Ž. a an asymptotic solution of the hybrid system of Riccati equations, associated with the singularly perturbed control problem with a small state delay; Ž b. conditions for the existence of a solution of this problem independent of the singular perturbation parameter ; Ž. c the design of a simplified controller with -independent gain matrices, which solves the problem for all sufficiently small. The approach proposed in this paper is valid for both standard and nonstandard forms of singularly perturbed delayed dynamics of the control problem. These forms are an extension of the ones considered for singularly perturbed dynamics without delay in 16.

3 SINGULARLY PERTURBED CONTROL PROBLEM 51 The paper is organized as follows. In Section 2, the control problem for a singularly perturbed linear system with a small state delay is formulated. The hybrid system of Riccati equations, associated with this problem, is written out. In Section 3, the formal zero-order asymptotic solution of this system of equations is constructed. Reduced-order Ž slow. and boundary-layer Ž fast. control problems, associated with the original one, are obtained, and their connection with the zero-order asymptotic solution is established. In Section 4, it is verified that the zero-order asymptotic solution is OŽ.-close to an exact solution. In Section 5, two controllers, solving the original problem, are obtained. In Section 6, an example illustrating the results of the previous sections is presented. In Appendix, the auxiliary lemma, applied in the verification of the zero-order asymptotic solution, is proved. Ž. n The following main notations are applied in this paper: 1 E is the n-dimensional real Euclidean space; Ž. 2 L b, c; E n 2 is the space of n-dimensional vector functions quadratically integrable on the interval Ž b, c.; Ž. 3 Cb, c; E n is the space of n-dimensional vector-functions continuous on the interval b, c ; Ž. 4 denotes the Euclidean norm either of the matrix or of the vector; Ž. 5 n L denotes the norm in L b, c; E 2 ; 2 Ž. n Ž. 4 n 6 C denotes the norm in Cb, c; E ; 7 col x, y, where x E, y E m, denotes the column block-vector with upper block x and lower block y; Ž. 8 I is the n-dimensional identity matrix; Ž. n 9 Re denotes the real part of a complex number ; Ž 1. xt Ž. dxž. t dt; Ž 11. x xt Ž. t, n where x E, t, b, Ž b.. Consider the system 2. PROBLEM FORMULATION xž t. A1xŽ t. A2 yž t. 1xŽ t h. 2 yž t h. B1uŽ t. Fw 1 Ž t., t, Ž 2.1. yž t. A3xŽ t. A4yŽ t. 3xŽ t h. 4yŽ t h. B2ut Fw 2 Ž t., t, Ž 2.2. Ž t. colc xž t. C yž t., už t. 4, t, Ž where x E n and y E m are state variables, u E r is a control, w E q p is a disturbance, E is an observation, A,, B, F, C Ž i i j j j i 1,...,4; j 1, 2. are constant matrices of the corresponding dimensions, isa small parameter Ž 1. and h is some constant.

4 52 GLIZER AND FRIDMAN Assuming that wt Ž. L, ; E q 2, we consider the performance index L2 L2 JŽ u, w. Ž t. wž t., Ž 2.4. where is a given constant. The control problem for a performance level is to find a controller u* xž., yž. that internally stabilizes the system Ž 2.1., Ž 2.2. and ensures the inequality Ju*, Ž w. for all wt L, ; E q and for xt Ž. 2, yt Ž., t. Consider the matrices A1 A2 1 2 A Ž,, 1. A Ž 1. A Ž 1. Ž 1. ž 3 4/ ž 3 4/ ž 2/ ž 2/ Ž 2.5a. F1 B 2 1 S FF BB, F, B, Ž 1. F Ž 1. B Ž 2.5b. D CC, C Ž C 1, C 2., Ž 2.5c. where the prime denotes the transposition. Consider the following hybrid system of matrix Riccati equations for P, QŽ. and RŽ,. in the domain Ž,. h, h,, PA A P PS P Q Q D, Ž 2.6. dqž. d A PS QŽ. RŽ,., QŽ h. P, Ž 2.7. Ž. RŽ,. QŽ. SQ Ž., R h, R, h Q. Ž 2.8. A solution of Ž 2.6. Ž 2.8. is a triple of Ž n m. Ž n m. -matrices P, QŽ., RŽ,.4, Ž,. h, h,, satisfying Ž 2.6. Ž 2.8., where QŽ. is continuously differentiable; RŽ,. is continuous; and RŽ,. and RŽ,. are piecewise continuous, while Ž. RŽ,. is continuous. Consider also the linear systems zž t. A B B P zž t. zž t h. h B B QŽ. zž t. d, t, Ž 2.9. zž t. A S PzŽ t. zž t h. S QŽ. zž t. d, t. Ž 2.1. h

5 SINGULARLY PERTURBED CONTROL PROBLEM 53 From 9 we obtain the following: if, for some, the problem Ž 2.6. Ž 2.8. has a solution PŽ., QŽ,., RŽ,,. such that the systems Ž 2.9. and Ž 2.1. with P PŽ., Q QŽ,. are asymptotically stable, then, for this, the controller Ž. Ž. Ž. Ž. h u* x, y t B P z t Q, z t d, z col x, y 4, Ž solves the control problem Ž 2.1. Ž The objectives of the present paper are: 1. To establish conditions Ž independent of. which ensure the existence of solution Ž of the control problem Ž 2.1. Ž 2.4. for all sufficiently small. 2. To derive a controller much simpler than Ž 2.11., which is constructed independently of and solves the control problem Ž 2.1. Ž 2.4. for all sufficiently small. The key point in reaching these objectives is the construction of a zero-order asymptotic solution to the problem Ž 2.6. Ž ZERO-ORDER ASYMPTOTIC SOLUTION OF TE PROBLEM Transformation of Ž 2.6. Ž 2.8. and Formal Zero-Order Asymptotic Solution Let us transform the problem Ž 2.6. Ž 2.8. to an explicit singular perturbation form. Following 1, we shall seek the solution of this problem in the form ž 2 3 / ž 3 4 / P1 P2Ž. Q1Ž,. Q2Ž,. P Ž., QŽ,., P Ž. P Ž. Q Ž,. Q Ž,. ž 2 3 / Ž 3.1. R1Ž,,. R2Ž,,. RŽ,,. Ž 1., Ž 3.2. R Ž,,. R Ž,,. where P Ž. and R Ž,,. Ž i 1, 2, 3. i i are matrices of the dimensions n n, n m, and m m respectively; Q Ž,. Ž j 1,..., 4. j are matri- ces of the dimensions n n, n m, m n, and m m respectively; P Ž. P Ž., R Ž,,. R Ž,,. Ž k 1, 3.. k k k k

6 54 GLIZER AND FRIDMAN Substituting Ž 2.5., Ž 3.1., and Ž 3.2. into the problem Ž 2.6. Ž 2.8., we obtain the following system in the domain Ž,. h, h, Žin this system, for simplicity we omit the designation of the dependence of the unknown matrices on.. P A P A A P A P PSP PS P1 PS P PSP Q Q D1, Ž 3.3. P A P A A P A P PSP PS P PS P PSPQ Q D2, Ž 3.4. P A P A A P A P 2 P SP PS P P S P PSP Q Q D3, Ž 3.5. dq d A PS PS 1 Ž Q1Ž. Ž A 3 PS 1 2 PS 2 3. Q3Ž. R1Ž,., Ž 3.6. dq d A PS PS 2 Ž Q2Ž. Ž A 3 PS 1 2 PS 2 3. Q4Ž. R2Ž,., Ž 3.7. dq d A P S PS 3 Ž Q1Ž. Ž A P S PS Q R Ž,., Ž 3.8. dq d A P S PS 4 Ž Q2Ž. Ž A P 4 2S2 PS 3 3. Q4Ž. R3Ž,., Ž 3.9. Ž. R, 2 Q SQ Q S Q1Ž. Q SQ Q Ž. SQ 3 3Ž., Ž 3.1. Ž. R, 2 Q SQ Q S Q2Ž. Q SQ Q Ž. SQ 3 4Ž., Ž Ž. R, 2 Q SQ Q S Q2Ž. Q SQ Q Ž. SQ 3 4Ž., Ž QkŽ h. P 1 k P2 k2 Ž k 1,2., l 2 l2 3 l Q h P P l 3,4, Ž R h, Q Q k 1,2, 3.14 k 1 k 3 k2

7 SINGULARLY PERTURBED CONTROL PROBLEM 55 R, h Q Q Ž. 4, R h, Q Q, Ž where S 2 FF BB, S 2 FF BB, S 2 F F B B, D C C, D C C, D C C 2. The problem Ž 3.3. Ž has the explicit singular perturbation form. Now, let us construct the zero-order asymptotic solution of this problem. Similarly to 1, we shall seek the zero-order asymptotic solution of the problem Ž 3.3. Ž in the form P i, QjŽ., RiŽ,.,, Ž i 1,2,3; j 1,...,4.. Ž Substituting 3.16 into and equating coefficients of in both parts of the resulting equations, we obtain the following system in the domain Ž,. h, h,. P A P A AP A P P S P P S P P S P P S P Q Q D, Ž P A P A A P P S P P S P Q Q 2 3 D2, Ž P A A P P S P Q Q D, Ž Ž. Ž. Ž dq d A P S P S Q R,, 3.2 dq d A P S P S Q R Ž,., Ž dq d A P S Q R,, 3.22 dq d A P S Q R Ž,., Ž Ž. R, Q 1 3 SQ 3 3Ž., Ž Ž. R, Q 2 3 SQ 3 4Ž., Ž Ž. R, Q 3 4 SQ 3 4Ž., Ž Qk Ž h. P1k P2 k2 Ž k 1,2., Ž QlŽ h. P3l Ž l 3, 4., Ž R h, Q k 1,2, 3.29 k 3 k2,

8 56 GLIZER AND FRIDMAN R, h Q 2 3 4, Ž 3.3. R h, 3 4Q4Ž.. Ž Remark 3.1. The problem Ž Ž can be divided into four simpler problems solved successively: Ž i. The First Problem consists of Ž 3.19., Ž 3.23., Ž 3.26., Ž 3.28., with l 4, and Ž Ž ii. The Second Problem consists of Ž 3.22., Ž 3.25., Ž 3.28., with l 3, Ž 3.29., with k 2, and Ž Ž iii. The Third Problem consists of Ž and Ž with k 1. Ž iv. The Fourth Problem consists of Ž 3.17., Ž 3.18., Ž 3.2., Ž 3.21., and Ž The First Problem and the Boundary-Layer Control Problem We assume that: A1. The First Problem has a solution P, Q Ž., R Ž,., Ž, h, h,, such that P P, R Ž,. R Ž, A2. All roots of the equation det I A S P expž h. S Q expž. d m h lie inside the left-hand half-plane. A3. All roots of the equation det I A B B P expž h. m lie inside the left-hand half-plane. LEMMA h 4 B B Q expž. d Under the assumptions A1A3, the matrix ž P3 Q4Ž. / Q4 R3Ž,. it is the kernel of linear bounded self-adjoint nonnegatie operator mapping the m space E L h,; E m into itself. 2 Proof. The statement of the lemma is a direct consequence of results of Ž 9 see Lemma 1 and its proof..

9 SINGULARLY PERTURBED CONTROL PROBLEM 57 The First Problem is the hybrid system of matrix Riccati equations associated with the control problem dy Ž. d A yž. yž h. 4 4 B u Fw Ž., ; Ž Ž y.,, L2 L2 J Ž u, w. Ž. w Ž., 4 Ž Ž. col C2 yž., u Ž.,, where y, u, w, and are state, control, disturbance, and observation respectively. In the following we shall call the problem Ž 3.32., Ž the boundary-layer Ž fast. problem associated with the original control problem Ž 2.1. Ž LEMMA 3.2. Under the assumptions A1A3, the controller u* y Ž.Ž. B P y Q yž. d soles the problem Ž 3.32., Ž h 4, q i.e., Ju*, w w L, ; E 2. Proof. The statement of the lemma directly follows from 9, Lemma The Second and the Third Problems LEMMA 3.3. The Second Problem and the Third Problem hae the unique solutions Q Ž., R Ž,.4 and R Ž,., respectiely, for Ž, h,h,. Moreoer, the matrices R Ž,. Ž k 1, 2. k hae the form R Ž,. Ž. where k k 3 3 k2, maxž h, h. Q Ž s. S Q Ž s. ds Ž k 1, 2., Ž Qk2,Ž h., h, kž. Ž 3.35 ½. Q 3 Ž h. k2, h, and Q 3 is a unique solution to the linear integral-differential equation dq d A P S Q Q Ž h h satisfying the initial condition 3.28 l 3. Q Ž s. SQ Ž s. ds Ž 3.36.

10 58 GLIZER AND FRIDMAN Proof. The lemma is an immediate consequence of the results of 1, Lemma The Fourth Problem and the Reduced-Order Control Problem Similarly to 1, pp , one can rewrite the Fourth Problem in the equivalent form where P A AP P SP D, Ž ž / h 3 P P N N Q d, Ž Qk P1k P2 k2 Ž k2, h A P S P S Q Ž s. ds k h R, s ds k 1, 2, 3.39 A A N A S N NS N, Ž S FFBB, F F1 NF, 1 2 B B1 NB, D D N A A N NSN, Ž N A SG M 1, N A 1 Ž Ž 3G D2. M 1, Ž M A4 SG, 3 G P3 Q4 d, h Ž Ai Ai i Ž i 1,...,4.. From the assumption A2 we directly obtain that the matrix M is invertible. In the following, we assume that: A4. The equation Ž has a symmetric positive semidefinite solution P 1. A5. All eigenvalues of the matrix Ž A SP. 1 lie inside the left-hand half-plane. A6. All eigenvalues of the matrix A Ž BB. P A B 1 lie 2 1 inside the left-hand half-plane, where FF N GM Ž A 2 2 A B B N., FF GM B B, M A B B G B

11 SINGULARLY PERTURBED CONTROL PROBLEM 59 From the assumption A3 we directly obtain that the matrix M is invertible. Now, let us present an interpretation of Eq. Ž and the assumptions A4A6. Setting in Ž 2.1. Ž 2.4., one obtains the control problem for the descriptor Ž algebraicdifferential. system Ez Ž t. A zž t. BuŽ t. FwŽ t., t, Ez, Ž L2 L2 J u, w t w t, t col Cz t, u t, t, Ž where z, u, w, and are state, control, disturbance, and observation, respectively, and ž / ž / ž / ž / I A 1 A 2 B1 F n 1 E, A, B, F. A 3 A4 B2 F2 Ž In the following, we shall call this problem the reduced-order Ž slow. one associated with the original control problem Ž 2.1. Ž Consider the generalized Riccati equation associated with the problem Ž 3.45., Ž K A A K KSK D, EK KE, Ž where S 2 FF BB. LEMMA 3.4. Under the assumptions A1, A2, A4, and A5, the matrix 1 / P1 K, ž G G where G P Q d Ž NP N., satisfies Ž h , and EK is positie semidefinite. Proof. The lemma is proved by direct substitution of K into Ž 3.48., applying the block expansion of Ž Žsee 27., and taking into account that G satisfies the Riccati equation G A A 4 4G GS3G D3 Žsee 1.. Note that 3.37 can be obtained from 3.48 by eliminating the lower left- and right-hand blocks of the matrix K of the dimensions m n and m m respectively.

12 6 LEMMA 3.5. GLIZER AND FRIDMAN Under the assumptions A1, A2, and A4, the system Ez Ž t. Ž A SK. zž t. Ž is asymptotically stable iff the assumption A5 is satisfied. Proof. Let x and y be the upper and lower blocks of the vector z of the dimensions n and m respectively. Since the matrix M is invertible Ž due to A2., one can rewrite Ž in the equivalent block form ž / ẋ t A SP x t, y t M A S P SG x t. Now, the statement of the lemma directly follows from A5. LEMMA 3.6. Under the assumption A1, A2, A3, and A4, the system Ž 3.5. Ez Ž t. Ž A BBK. zž t. Ž is asymptotically stable iff the assumption A6 is satisfied. Proof. The lemma is proved similarly to Lemma 3.5. LEMMA 3.7. Under the assumptions A1A6, the controller u* zt Ž. BK zž t. soles the problem Ž 3.45., Ž 3.46., i.e., Ju*, Ž w. wž t. L, ; E q. 2 Proof. The lemma is a direct consequence of Lemmas , and it is proved similarly to 9, Lemma 1, applying the functional VŽ z. zek z. Thus, we have shown that Eq. Ž and the assumptions A4A6 are associated with the reduced-order control problem Ž 3.45., Ž by conditions of the existence of its solution. We have completed the construction of the zero-order asymptotic solution to the problem Ž 3.3. Ž and, hence, to Ž 2.6. Ž It is clear that the asymptotic approach to the problem Ž 2.6. Ž 2.8. essentially simplifies a procedure of its solution. The original problem is reduced to three problems of lower dimensions solved successively. These problems are the First Problem, the problem Ž 3.36., Ž Ž l 3., and the equation Ž Note that these problems are independent of. The other components of the zero-order asymptotic solution are obtained from the explicit expressions Ž 3.34., Ž 3.38., Ž In the next section, we shall verify the zero-order asymptotic solution to the problem Ž 3.3. Ž constructed in this section.

13 SINGULARLY PERTURBED CONTROL PROBLEM VERIFICATION OF TE ZERO-ORDER ASYMPTOTIC SOLUTION OF TE PROBLEM Auxiliary Results In this subsection, we shall present some auxiliary results which will be applied in the verification of the zero-order asymptotic solution to the problem Ž 3.3. Ž Consider the system Ž t. A Ž t. Ž t h. G Ž,. Ž t. d, where h ž / ž Ž. Ž. / Ž 1. Ž. Ž 1. Ž. ž G Ž,. G Ž,. / Ž 1. G Ž,. Ž 1. G Ž,. A A 1 2Ž. ÃŽ., Ž 1. A Ž. Ž 1. A Ž Ž., t, E nm, Ž 4.1. Ž 4.2. G Ž,., Ž the blocks A Ž., Ž., G Ž, are of dimension n n, and the blocks A Ž., Ž., G Ž, are of dimension m m. We assume that: A7. A Ž., Ž., and G Ž,. Ž i 1,..., 4. i i i are differentiable func- tions of and Ž,. for h, and all sufficiently small. A8. The reduced-order subsystem associated with Ž 4.1., where n 1 t 1 t, t, 1 E, , Ž 4.5. A G Ž,. d Ž i 1,...,4. i i i i h is asymptotically stable.

14 62 GLIZER AND FRIDMAN A9. The boundary-layer subsystem associated with 4.1, d Ž. d A Ž. Ž h m G 4, 2 d,, 2 E, 4.6 h is asymptotically stable. Let Ž t,. be the fundamental matrix of the system Ž 4.1., i.e., it satisfies this system and the initial conditions Ž,. I nm; Ž t,., t. Ž 4.7. LEMMA 4.1. Let Ž t,., Ž t,., Ž t,., and Ž t, be the upper left-hand, upper right-hand, lower left-hand, and lower right-hand blocks of the matrix Ž t,. of the dimensions n n, n m, m n, and m m respectiely. Under the assumptions A7A9, for all t and sufficiently small, the following inequalities are satisfied: kž t,. a expž t. Ž k 1,3., 2Ž t,. a expž t., 4Ž t,. a expž t. expž t., where a,, and are some constants independent of. For a proof of the lemma, see Appendix. Consider the particular case of the system Ž 4.1. with the coefficients A Ž. A S P Ž., Ž., G Ž,. SQ, Ž 4.8. where / ž / ž S P S P / Ž 1. S P Ž 1. S P P1 P2 Q1 Q2 PŽ., Q, Ž 4.9. P P ž Q Ž. Q Ž Ž 4.1. Let Ž t,. be the fundamental matrix of the system Ž 4.1., Ž Let Ž t,., Ž t,., Ž t,., and Ž t,. be the upper left-hand, upper

15 SINGULARLY PERTURBED CONTROL PROBLEM 63 right-hand, lower left-hand, and lower right-hand blocks of the matrix Ž t,. of the dimensions n n, n m, m n, and m m respectively. LEMMA 4.2. Under the assumptions A1, A2, A4, and A5, the inequalities k Ž t,. a expž t. Ž k 1,3., 2 Ž t,. a expž t., 4Ž t,. a expž t. expž t., are satisfied for all t and sufficiently small ; where a,, and are some constants independent of. Proof. Let us construct the reduced-order and the boundary-layer subsystems, associated with the system Ž 4.1., Ž 4.8., and show the asymptotic stability of these subsystems. From Ž 4.2., Ž 4.3., and Ž 4.8. one has A A SP S P, A Ž. A2 S1P2 S2P 3, Ž A A S P S P, A Ž. A4 S2P2 S3P 3, Ž G kž,. S1Qk SQ 2 k2, S2 P2 k Ž k 1,2., Ž G l Ž,. S2 Ql2, SQ 3 l S3P2l2 Ž l 3,4.. Ž The block representation of the matrix is given in Ž 2.5a.. Substituting Ž Ž into Ž 4.5., one obtains the matrix of coefficients of the reduced-order subsystem Ž 4.4. associated with the system Ž 4.1., Ž 4.8., Ž A SP SG N A S P SG. Ž Under the assumption A2, the matrix M in the expression for N1 is invertible. Substituting the expression for G Ž see Lemma 3.4. into Ž yields, after some rearrangement, A SP 1, which implies, along with the assumption A5, the asymptotic stability of the reduced-order subsystem, associated with the system Ž 4.1., Ž Replacing in Ž 4.6. A Ž. with its expression from Ž 4.12., Ž. 4 4 with 4, and G Ž,. with its expression from Ž 4.14., we obtain the boundary-layer 4

16 64 GLIZER AND FRIDMAN subsystem, associated with the system 4.1, 4.8 : d Ž. d A S P Ž. Ž h h SQ Ž. d,, 2 E m. Ž The assumption A2 directly implies the asymptotic stability of the boundary-layer subsystem Ž Now, the statement of the lemma is an immediate consequence of Lemma Estimation of the Remainder Term Corresponding to the Zero-Order Asymptotic Solution TEOREM 4.1. Under the assumptions A1, A2, A4, and A5, the problem Ž 3.3. Ž has a solution P Ž., Q Ž,., R Ž,,. Ž i j i i 1, 2, 3; j 1,...,4. for all sufficiently small, and this solution satisfies the inequalities PiŽ. Pi a, QjŽ,. QjŽ. a, R Ž,,. R Ž,. a, i i where Ž,. h, h,; P, Q and R Ž,. i j i are defined in Section 3; and a is some constant independent of. Proof. Let us transform the variables in the problem Ž 3.3. Ž as PiŽ. Pi Pi Ž i 1,2,3., Ž QkŽ,. QkŽ. QkŽ,., l l 2 l2 Ql Q, Q P,, 4.18 Ž k 1,2; l 3,4., R Ž,,. R Ž,. Q Ž. P R2Ž,,. R2Ž,.,,, 4.19 R1 ½ Q P Q h Q,,, 4.2 R2 R Ž,,. R Ž,. Q Ž. P ,, R3

17 SINGULARLY PERTURBED CONTROL PROBLEM 65 The transformation Ž Ž yields the following problem for the new variables Ž., Ž,., and Ž,,. Ž i 1, 2, 3; j 1,..., 4. Pi Qj Ri in the domain Ž,. h,h,. A Ž. A Ž,. Ž,. P P Q Q DPŽ. P S P, Ž d Ž,. d A Ž,. Ž 1. G Ž,. Q Q P RŽ,,. DQŽ,. P S QŽ,., Ž Ž. R Ž,,. Ž 1. Ž,. G Ž,. Ž 1. G Ž,. Ž,. Q D,, R QŽ,. S QŽ,., Ž QŽ h,. P, R R Q h,,, h,,, where A Ž. and G Ž,. are defined by Ž Also in Ž Ž 4.25., P 2 P 3 / P1 P 2 P, ž Ž. Ž. ž / Q1Ž,. Q2Ž,. QŽ,., Ž,. Ž,. Q3 R1Ž,,. R2Ž,,. RŽ,,. Ž 1., ž Ž,,. Ž,,. Q4 R2 R3 / DP 2Ž. D D / P 2 P 3 D Ž., P ž ž / DQ1Ž,. DQ2 Ž,. DQŽ,., D Ž,. D Ž,. Q3 DR1Ž,,. DR2Ž,,. DR Ž,,.. ž D Ž,,. D Ž,,. Q4 R2 R3 / Q Ž The matrices D Ž., D Ž,., D Ž,,. Pk Qj Ri are known functions of P i, Q Ž., and R Ž,. Ž k 2, 3; i 1, 2, 3; j 1,..., 4.. These j i

18 66 GLIZER AND FRIDMAN matrices are continuous in, h, h,, and for all sufficiently small they satisfy the inequalities DPkŽ. a, DQjŽ,. a, DRiŽ,,. a Ž k 2,3; j 1,...,4; i 1,2,3., Ž where a is some constant independent of. Denote PŽ P. DPŽ. P S P Ž., Ž QŽ P, Q.Ž,. DQŽ,. P S QŽ,., Ž 4.28.,, D,, RŽ Q. R QŽ,. S QŽ,., Ž Ž t,,. Ž t h,. h Ž 1. Ž t,. G Ž,. d. Ž 4.3. Applying Lemma 4.2, one can directly show that the matrix Ž t,,. satisfies the inequalities for all t, h,, and sufficiently small, kž t,,. a expž t., l Ž t,,. a expž t. 1 Ž 1. expž t., Ž where Ž k 1, 2; l 3, 4. Ž t,,. Ž j 1,..., 4. j are the upper left-hand, upper right-hand, lower left-hand, and lower right-hand blocks of this matrix of the dimensions n n, n m, m n, and m m respectively; a,, and are some constants independent of. Applying results of 1, pp. 5152, we can rewrite the problem Ž Ž in the equivalent form P PŽ P. t, t, Q P Q h Ž t,., Ž,. Ž t,. d Q P Q h Ž t,., Ž,. Ž t,. d R Q h h Ž t,. Ž,,. Ž t,. d d dt, Ž 4.32.

19 SINGULARLY PERTURBED CONTROL PROBLEM 67 Ž,. Ž t,. Ž. Ž t,,. Q P P R Ž,,. Q P Q h Ž t,. Ž,. Ž,. Ž t,,. d Q P Q h Ž t,., Ž,. Ž t,,. d RŽ Q. 1 h h Ž t,. Ž,,. Ž t,,. d d dt 1 1 h Q P Q Ž t,., Ž t,. R Q h Ž t,. Ž, t,. d dt, Ž P P Ž t,,. Ž. Ž t,,. QŽ P Q. 1 Ž 1. 1 h t,,,, t, d 1 QŽ P Q. 1 1 h t,,,, t,, d 1 RŽ Q. 1 2 h h Ž t,,. Ž,,. Ž t,,. d d dt h Q P Q Ž,.Ž t,. Ž t,,. R Q 1 Ž 1. 1 h, t, t,, d dt h Q P Q Ž t,,., Ž t,. Ž 1. RŽ Q. Ž 1. 1 h t,,, t, d dt min Ž h, h. R Q Ž.Ž t, t,. dt. Ž 4.34.

20 68 GLIZER AND FRIDMAN It is obvious that minž h, h. h Ž,. h, h,. Ž Now, applying the procedure of successive approximations to the system Ž Ž and taking into account Lemma 4.2, the equations Ž Ž and the inequalities Ž 4.26., Ž 4.31., Ž 4.35., one directly obtains the existence of the solution Ž., Ž,., Ž,,. P Q R of the problem Ž Ž Žand, consequently, of the problem Ž Ž , satisfying the inequalities for all sufficiently small and Ž,. h, h,, Pi a, QjŽ,. a, RiŽ,,. a Ž i 1,2,3; j 1,...,4., Ž where a is some constant independent of. The inequalities Ž along with the equations Ž Ž immediately yield the statements of the theorem. COROLLARY 4.1. Under the assumptions A1, A2, A4, and A5, the system Ž 2.1., where P PŽ. and QŽ. QŽ,. are defined in Theorem 4.1, is asymptotically stable for all sufficiently small. Proof. The corollary is an immediate consequence of Theorem 4.1 and Lemma 4.1. It is proved similarly to Lemma CONTROLLERS FOR PROBLEM Consider the controller u xž., yž.ž t. of the form Ž 2.11., where PŽ. and QŽ,. are defined by Ž 3.1., and P Ž. and Q Ž,. Ž i j i 1, 2, 3; j 1,..., 4. are components of the solution to the problem Ž 3.3. Ž mentioned in Theorem 4.1. Consider also the following controller with -independent gain matrices: Ž. Ž. Ž Ž. u x, y t BP BG x t 2 3 h 4 B P yž t. Q yž t. d. Ž 5.1.

21 SINGULARLY PERTURBED CONTROL PROBLEM 69 LEMMA 5.1. Under the assumptions A1A6, the controller u xž., yž.ž t. internally stabilizes the system Ž 2.1., Ž 2.2. for all sufficiently small. Proof. Substituting Ž 5.1. into Ž 2.1., Ž 2.2. and setting wt, we obtain the system where xž t. Aˆ xž t. Aˆ yž t. xž t h. yž t h ˆ1 h G y t d, t, 5.2 yž t. Aˆ xž t. Aˆ yž t. xž t h. yž t h ˆ2 h G y t d, t, 5.3 Aˆ A BBP BBG, Aˆ A BBP, Ĝ BBQ, Aˆ A B B P B B G, Aˆ A B B P, Ĝ B B Q. Ž 5.4. Ž 5.5. Thus, in order to prove the lemma, one has to prove the asymptotic stability of the system Ž 5.2., Ž 5.3. for all sufficiently small. Let us show the asymptotic stability of the reduced-order and the boundary-layer subsystems, associated with the system Ž 5.2., Ž Setting in Ž 5.2., Ž 5.3., we obtain a system, coinciding with the system Ž ence, due to Lemma 3.6, the reduced-order subsystem, associated with Ž 5.2., Ž 5.3., is asymptotically stable. The asymptotic stability of the boundary-layer subsystem, associated with Ž 5.2., Ž 5.3., is an immediate consequence of the assumption A3. ence, by Lemma 4.1, the system Ž 5.2., Ž 5.3. is asymptotically stable for all sufficiently small. From Theorem 4.1 and Lemma 5.1 we obtain the following corollary: COROLLARY 5.1. Under the assumptions A1A6, the controller u xž., yž.ž t. internally stabilizes the system Ž 2.1., Ž 2.2. for all sufficiently small. TEOREM 5.1. Under the assumptions A1A6, the controller u xž., yž.ž t. soles the control problem Ž 2.1. Ž 2.4. for all sufficiently small.

22 7 GLIZER AND FRIDMAN Proof. For all sufficiently small, the theorem follows from 9 and Corollaries 4.1 and 5.1. TEOREM 5.2. Under the assumptions A1A6, the controller u xž., yž.ž t. soles the control problem Ž 2.1. Ž 2.4. for all sufficiently small. Ž. Ž. Proof. Substituting u x, y t into , one has xž t. Aˆ xž t. Aˆ yž t. xž t h. yž t h ˆ1 1 h G Ž. yž t. d FwŽ t., Ž 5.6. yž t. Aˆ xž t. Aˆ yž t. xž t h. yž t h ˆ2 2 h G Ž. yž t. d FwŽ t., Ž 5.7. Ž t. col CzŽ t., u xž., y Ž t., t, Ž L 2 L 2 J w t w t ˆ ˆ 1 P1 2 P 2 xž t. D D xž t. 2 xž t. D D yž t. Ž 3 P 3. yž t. D Dˆ yž t. 2 xž t. Dˆ Ž,. yž t. d h Q1 2 yž t. Dˆ Ž,. yž t. d h Q2 ˆR1 h h yž t. D Ž,,. yž t. d d dt 2 w t w t dt, 5.9 where ˆD P1 P1B1 GB 1 2 BP 1 1 BG 2 1, ˆD P 2 P1B1 GB 1 2 BP 2 3, Ž 5.1.

23 SINGULARLY PERTURBED CONTROL PROBLEM 71 Dˆ P B B P, Dˆ Ž,. Ž 1. P B GB BQ Ž., P Q ˆD Q2Ž,. Ž 1. P3B2B2Q4Ž., ˆ 2 R D Ž,,. 1 Q Ž. B B Q Ž.. Ž Ž Note that, according to Lemma 5.1, the system 5.6, 5.7 is internally asymptotically stable for all sufficiently small. Thus, in order to prove the theorem, we have to show that for all sufficiently small Consider the block matrices J Ž w. for all wž t. L,; E q 2 and for xž t., yž t., t. Ž ž / ž / Aˆ Aˆ 1 Gˆ 1 2 1Ž. Aˆ, Gˆ Ž,., ˆ ˆ 2 Ž 1. A Ž 1. A Ž 1. Gˆ Ž ž / Dˆ Dˆ Dˆ Q1Ž,. P1 P 2 Dˆ, Dˆ P QŽ,., Dˆ Dˆ Dˆ, P 2 P 3 Q2 ž R1/ ˆD R Ž,,. ˆ, D,, Ž Ž and the hybrid system of matrix Riccati equations for P, ˆ QˆŽ., and RˆŽ,. in the domain Ž,. h,h,, PA ˆˆ Aˆ P ˆ PS ˆˆ Pˆ Qˆ Q ˆ D Dˆ, Ž P ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ Q dqž. d A PS QŽ. PGŽ,. RŽ,. D Ž,., ˆ ˆ ˆ ˆ ˆ R, G, Q Q G, Ž Q ˆ Ž. SQ ˆ ˆŽ. Dˆ Ž,,., Ž QˆŽ h. P ˆ, RˆŽ h,. R ˆ Ž, h. Q ˆ, Ž ˆ 2 FF. where S R

24 72 GLIZER AND FRIDMAN Consider also the system zž t. Aˆ Sˆ Pˆ zž t. zž t h. ˆˆ SQ GˆŽ,. zž t. d, t. Ž 5.2. h Similarly to 9, one can obtain the following: if for some, such that Ž 5.6., Ž 5.7. is internally asymptotically stable, the problem Ž Ž has a solution PˆŽ., QˆŽ,., RˆŽ,,., and the system Ž 5.2. with Pˆ PˆŽ., QˆŽ. QˆŽ,. is asymptotically stable, then the inequality Ž is satisfied for this. We shall seek the solution of the problem Ž Ž in the form ž / ž / Pˆ Pˆ Qˆ, Qˆ Ž,. PˆŽ., QˆŽ,., Pˆ Pˆ Ž. Qˆ Ž,. Qˆ Ž, ž / Ž Rˆ,, Rˆ 1 2Ž,,. ˆR Ž,,. Ž 1. ˆ, Ž R Ž,,. Rˆ Ž,,. 2 3 where the matrices PˆŽ., QˆŽ,., and RˆŽ,, are of the dimension n n; the matrices PˆŽ., Qˆ Ž,., and Rˆ Ž,, are of the dimension m m; PˆŽ. Pˆ Ž., and R Ž,,. Rˆ Ž,,. Ž k 1, 3. k k k k. Similarly to Theorem 4.1, it can be verified that, for all sufficiently small, the problem Ž Ž has the solution in the form Ž 5.21., Ž satisfying the inequalities PˆŽ. P a, Qˆ Ž,. Q Ž. a, i i j j ˆR Ž,,. R Ž,. a, i i where Ž,. h,h, Ž i 1, 2, 3; j 1,...,4.; P, Q i j and R Ž,. i are defined in Section 3; and a is some constant independent of. Now, similarly to Corollary 4.1, we have that the system Ž 5.2. with Pˆ PˆŽ., QˆŽ. QˆŽ,. is asymptotically stable for all sufficiently small which completes the proof of theorem.

25 SINGULARLY PERTURBED CONTROL PROBLEM EXAMPLE Consider an example of the problem with the following data: n m r q 1 and A 3, A 1, A 1, A 2, 2, 1, 1, 1, B1 6, B2 1, F1 2, F2.5, C1 2, C2 1,.5. Ž 6.2. In order to save the space, we do not rewrite the problem Ž 3.3. Ž with the data Ž 6.1., Ž Applying the results of Section 3, we shall construct the asymptotic solution to this problem. Under Eqs. Ž 6.1., Ž 6.2., the First Problem Ž see Remark 3.1. becomes 4P 2Q 1, dq4 d 2Q4 R3Ž,., h,, Q Ž h. P, Ž Ž. R3 Ž,., Ž,. h, h,, R Ž h,. Q Ž.. Ž Solving Ž 6.5., one directly has Q4 Ž h., if h, R3 Ž,. Ž 6.6 ½. Q 4 Ž h., if h. Substituting Ž 6.6. into Ž 6.4., we obtain the functional-differential equation for Q Ž., 4 dq4 d 2Q4 Q4Ž h., This equation has a unique solution h,, Q Ž h. P. Ž Q4 P3 Ž., f Ž h. expž ' 3. f Ž h. expž ' 3. f Ž h., h,, 1 2 ' ' ' ' ' ' ' ' Ž 6.8. where f Ž h. 3 2 expž 3 h., f Ž h. 3 2 expž 3 h. 1 2, and f Ž h. Ž 2 3. expž 3 h. Ž 2 3. expž 3 h.. It is obvious that f h h k,1, k

26 74 GLIZER AND FRIDMAN Standard analysis of the function yields max Ž h. 1 h. Ž 6.1. h, Substituting Ž 6.8. into Ž 6.3. and solving the resulting equation with respect to P, we have 3 P3 fž h. fž h., Ž where fž h. 4' 3 Ž 6 4' 3. expž ' 3 h. Ž 6 4' 3. expž ' 3 h.. Some easy analysis shows that fž h. h and, therefore, Moreover, it can be shown that P3 h. Ž max P P.5. Ž h 3 3 h Using 6.8, 6.9, and 6.12, one directly has Q4, h,, h. Ž Let us show that the assumptions A2 and A3 are satisfied. Begin with A2. Using the data Ž 6.1., Ž 6.2., we obtain the equation in A2, 1 2 expž h.. Further, we have for any h, Re 1 2 Re Re expž h. 1 Re 1 : Re. Ž ence, all roots of the equation 1 lie inside the left-hand half-plane for all h. Now, let us proceed to A3. The equation in A3 is 2 P expž h. Q expž. d h Ž Taking into account 6.8, 6.1, 6.12, and 6.14, we have from 6.16 Re 2 2 P3 Re Re expž h. 4 h Q Re exp d 1 1 h P Re : Re

27 SINGULARLY PERTURBED CONTROL PROBLEM 75 Consider the inequality with respect to h, 1 Ž 1 h. P3. Ž The solution of this inequality is h Ž Now, using Ž Ž 6.19., one has that all roots of the equation Ž lie inside the left-hand half-plane for all h satisfying Ž ence, the assumption A3 is satisfied for all h satisfying Ž Proceed to the Second and the Third Problems Ž see Remark In order to obtain the solutions to these problems, one has Žaccording to Lemma 3.3. to solve the equation Ž with the initial condition Ž Ž l 3.. Under the data Ž 6.1., Ž 6.2., the problem Ž 3.36., Ž Ž l 3. becomes dq3 d 2Q3 Q4Ž h., Solving 6.2, we obtain where h,, Q Ž h. P. Ž Ž '. Ž '. Q P f h exp 3 f h exp 3 f h, Ž '. Ž '. Ž '. Ž '. f h f h exp 3 h 2 3, 3 2 f h f h exp 3 h h,, Ž Using Ž 3.34., Ž 3.35., we obtain for k 1, 2 ½ Qk2,Ž h., if h, Rk Ž,. k Ž Ž 1. Q Ž h., if h, 3 Now, let us proceed to the Fourth Problem Ž see Remark In Section 3, this problem has been reduced to the equations Ž Ž In order to solve these equations, we have to calculate the matrices defined in Ž 3.4. Ž Under the data Ž 6.1., Ž 6.2., these matrices become N1 2Ž G 1., N22, F 3 G, B 24 Ž G., Ž A 1, S 42G Ž 7., D 4, Ž where G is defined in It is clear that G h.

28 76 GLIZER AND FRIDMAN Using Ž 6.8., Ž 6.1., and Ž Ž 6.14., one has for all h, satisfying Ž 6.19., G Ž 1 hp This inequality along with the expression for S Žsee Ž yields S h, Ž Substituting 6.24 into 3.37, we obtain after some rearrangement 2 22G7 P1 P The inequality Ž implies that Ž has the single positive solution for all h, satisfying Ž ' P G Ž Ž 2G.. Ž By direct calculation we have A SP h, Ž ence, the assumption A5 is satisfied for all such h. Now, let us verify that the assumption A6 holds. Calculating the matrices and, one has A B A 4Ž G 3. Ž G 1., B4Ž G 3.Ž G 4. GŽ G 1.. Ž Substitution A 1, B 24 G, and 6.29 into the matrix of A6 yields A A BBB P1 5G 11 16Ž G 4. P1 Ž G 1.. Ž 6.3. Some easy analysis shows that the expression in the right-hand part of Ž 6.3. is negative for all h, satisfying Ž ence, the assumption A6 is satisfied for all h, satisfying Ž Substituting the expressions for N and N Žsee Ž into Ž yields 1 2 P 21 Ž G. P Q d 2. Ž h

29 SINGULARLY PERTURBED CONTROL PROBLEM 77 Using 6.1, 6.2 and 6.22, 6.31, one obtains from 3.39 Q 2GP Q Ž s. ds P Q Ž s. ds h h 3 h Q s h ds, 6.32 Q Ž 3 2G. P Q Ž s. ds P Q Ž s. ds h h 3 h Q s h ds Thus, we have completed the construction of the zero-order asymptotic solution to the problem Ž 3.3. Ž for the data Ž 6.1., Ž aving this asymptotic solution, one can construct the controller u xž., yž., given by Ž 5.1., which solves the control problem Ž 2.1. Ž 2.4. for all sufficiently small. Giving any value of h, satisfying Ž 6.19., one can obtain the numerical expression for u xž., yž.ž t.. Thus, for h.4, we find u xž., yž. Ž. t xž t..378 yž t. Ž '. Ž '..893 exp exp 3 yž t. d..4 For h.6, we find u xž., yž. Ž. t xž t yž t. Ž '. Ž '. Ž exp exp 3 yž t. d..6 Ž APPENDIX: PROOF OF LEMMA 4.1 Proof of Lemma 4.1 is based on the decoupling transformation of a singularly perturbed system with a small delay Ž Such a transformation was introduced in 8 in the case when, G Ž k 1, 3. k k. In Subsection A.1 we will generalize the results of 8 to the case of nonzero, G Ž k 1, 3.. k k

30 78 GLIZER AND FRIDMAN A.1. Slowfast Decomposition of System Ž 4.1. For each denote by TŽ. t : C h,; E n C h,; E n and St, Ž.: C h,; E m C h,; E m, t, the semigroups of the solution operators, corresponding to the linear equations and xž t., t, xž. x Ž., h,, Ž A.1. yž t. A y, A y b t b t A yž t. yž t h h G, y t d, t ; yž. y Ž., h,, Ž A.2. defined by TŽ t. x Ž. xž t. and SŽ t,. y Ž. yž t., h,, where t is considered as a parameter and x Ž. and y Ž. are continuous for h,. Let Yt, Ž., t h,. be the fundamental matrix of Ž A.2., Y I m; Y Ž.,. Denote St, Ž. Y Ž. Yt Ž,.Žt, h,.. Let XŽ. t I, t, XŽ. t, t, and TŽ. t X Ž. XŽ t.ž n t, h,., where X I, X Ž. n,. Introduce the new variable z x xž t., z Q C h,; E n : 4 t. t t t Evidently Q is invariant with respect to TŽ. t. Under the assumption A7, we can represent the right-hand part of Ž 4.1., where Ž t. col xt Ž., yt Ž.4, in the form ž / ž / A11Ž. A12Ž. xt, Ž A.3 1 A y y. Ž 1. A Ž. A Ž. t b t ž / where A Ž. Ž i 1, 2, j 1, 2. are linear operators on C h,; E n ij and C h,; E m. Applying the variation of constants formula 14 to Ž 4.1. with the initial condition col x Ž., y Ž.4, we obtain the equivalent system of differential

31 SINGULARLY PERTURBED CONTROL PROBLEM 79 and integral equations with respect to xt, Ž. z, and y, xž t. A Ž. xž t. z A Ž. y, x xž t. z, 11 t 12 t t t t z TŽ t. z TŽ t s.ž X I. t n A Ž. xž s. z A Ž. y ds, t 11 s 12 s A.4 y SŽ t,. y Ž 1. SŽ t s,. Y t A21Ž. x s A22Ž. ys ds, t t where z z t t. Note that Ž 4.6. corresponds to Ž A.2. written in the fast time t. Then the asymptotic stability of Ž 4.6. implies the following inequality for all t and sufficiently small : SŽ t,. y a expž t. y C, C sup SŽ t,. Y a expž t., h, a,. Ž A.5. Since TŽ. t z and TŽ.Ž t X I. n for t h, and z Q, we have for all t and sufficiently small TŽ t. z a expž t. z C, C sup TŽ t.ž X I. a expž t., h, n a,. Ž A.6. By a standard argument for the existence of invariant manifolds Žsee e.g. 14, 6., the system Ž A.4. has the center manifold for all sufficiently small, zt L1Ž. xž t., yt L2Ž. xž t., Ž A.7. n n where L : E Q, L : E C h,; E m 1 2 are linear bounded operators. The flow on the center manifold is governed by the equation ẋž t. A Ž. I L Ž. A Ž. L Ž. xž t.. Ž A n

32 8 GLIZER AND FRIDMAN For continuously differentiable functions C h,; E n, C h,; E m, denote, if h,., A ½, if, ½, if h,., B Ž 1. Ab, if. The latter operators are extensions of infinitesimal generators of the semigroups TŽ. t and St, Ž. to the space of continuously differentiable functions 14. Similarly to 6, the following proposition can be proved: PROPOSITION A.1. Under the assumptions A7 and A9, for all sufficiently small : 1. the continuously differentiable in h, Ž n n. - and Ž m n. -matrix functions L Ž. L Ž,. and L Ž. L Ž, , such that L Ž,., determine the center manifold Ž A.7. 1 iff for eery h, they satisfy the equation L 1Ž. A11Ž. Ž In L1Ž.. A12Ž. L2Ž. ž L2Ž./ ž A L Ž. Ž X I. A Ž. Ž I L Ž.. 1 n 11 n 1 A12Ž. L2Ž. / BŽ. L2Ž. Ž 1. Y A21Ž. Ž In L1Ž.. A22Ž. L2Ž. Ž A.9. where 2. the matrix, defined in 4.5, is nonsingular and the approximation 4 3 ž / ž / L1Ž. 1 L OŽ., L2 4 3, Ž A.1. L2Ž. 2 is gien in Ž 4.5., holds for all h,. Changing the variables in A.4 t zt L1Ž. xž t. Ž t Q., t yt L2Ž. xž t., and using results of 14, Eq. 4.8, we obtain the system xž t. A Ž. I L Ž. A Ž. L Ž. xž t. 11 n A A, A t 12 t

33 SINGULARLY PERTURBED CONTROL PROBLEM 81 t TŽ t. TŽ t s. L Ž. Ž I X. t 1 n A11 s A12 s ds, Ž A.12. t SŽ t,. Ž 1. SŽ t s,. L Ž. A A t 2 11 s 12 s Y A A ds, Ž A s 22 s where, t t t t. From Ž A.5. and Ž A.6. one can derive the following exponential bounds on the solutions to Ž A.12. and Ž A.13. for all t and sufficiently small : a exp t t C t C C C, a,. Ž A.14. Similarly to 6, one can show that the system Ž A.11. Ž A.13. has the stable manifold for all sufficiently small, xž t. M1 t M2 t, Ž A.15. n m n where M1 : Q E, M2 : C h,; E E are linear bounded Ž uniformly in. operators. Similarly to 8, we can show that after the following change of variables xt Ž. xt Ž. M M 1 t 2 t we obtain the decoupled system of Ž A.8. and Ž A.12., Ž A.13.. Expressing xž t,. z t, and y by xt, Ž., and, we obtain the following lemma. t t t LEMMA A.1. Under the assumptions A7 and A9, for all sufficiently small n there exists an inertible operator T : E Q C h,; E m n E Q C h,; E m, gien by 4 4 col xž t., z, y TŽ. col xž t.,,, t t t t In M1 M2Ž. TŽ. L1Ž. In L1Ž. M1 L1Ž. M2Ž., Ž A.16. L L Ž. M Ž. I L Ž. M Ž. and T 1 Ž m 2 2 In M1Ž. L1Ž. M2Ž. L2Ž. M1Ž. M2Ž. L1Ž. In L Ž. I 2 m 4,

34 82 GLIZER AND FRIDMAN which transforms Ž A.4. to the purely slow system Ž A.8. and the purely fast system of Ž A.12. and Ž A.13.. ere, In and Im denote the identity operators on the corresponding spaces. A.2. Proof of Lemma 4.1 From Ž A.1. it follows that Ž A.8. can be rewritten in the form ẋž t. OŽ. xž t., where is given by Ž Therefore, under the assumption A8, the solution of Ž A.8. satisfies the following inequality for all t and sufficiently small, xž t. a expž t. xž., a,. Ž A.17. Lemma A.1 and the inequalities Ž A.14. and Ž A.17. imply that the solution nm to 4.1 with the initial condition C h,; E satisfies the following inequality for all t and sufficiently small : Ž t. a expž t., a,. The latter inequality immediately implies the exponential bound for the fundamental matrix t, for all t and sufficiently small, Ž t,. a expž t., a,. Ž A.18. Thus the inequalities for 1 and 3 of Lemma 4.1 are satisfied. Now, let us prove the inequalities for 2 and 4. Denoting ž 2Ž t, Ž t,.. / 4Ž t,. and using 4.1, 4.7, we obtain the equation for t, dž t,. dt A Ž. Ž t,. Ž. Ž t h,. and the initial conditions GŽ,. Ž t,. d, t, Ž A.19. h ž m / Ž,. I ; Ž,.,. Ž A.2.

35 SINGULARLY PERTURBED CONTROL PROBLEM 83 Let the m m -matrix t, satisfy the equation dž t,. dt A Ž. Ž t,. Ž. Ž t h, h G, t, d, t, A.21 and the initial conditions Ž,. I m; Ž,.,. Ž A.22. Consider the equation det I A Ž. Ž. expž h. G Ž,. expž. d. m h Ž A.23. Taking into account the assumptions A7 and A9, and applying results of 1, Proposition 4.3, one obtains that all roots of Ž A.23. lie inside the left-hand half-plane for all sufficiently small. Moreover, similarly to this result of 1, it can be shown that Re 2, Ž A.24. for all sufficiently small. Assumption A7 and Ž A.24. yield the following inequality for the solution to Ž A.21., Ž A.22. for all t and sufficiently small : Ž t,. a expž t., a,. Ž A.25. Changing the variable in A.19, A.2 as ž / Ž t,. Ž t,. Ž t,., Ž t,., Ž A.26. Ž t,. we obtain the problem where d Ž t,. dt A Ž. Ž t,. Ž. Ž t h,. G Ž,. Ž t,. d Ž t,., Ž A.27. h Ž,.,, Ž A.28. Ž t,. A Ž. Ž t,. Ž. Ž t h,. G Ž,. Ž t,. 2 2 h 2 d. ž /

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