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A New Approach to Lipschitz Continuity in State Constrained Optimal Control1

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A New Approach to Lipschitz Continuity in State Constrained Optimal Control1 Systems & Control Letters 35 (1998) 137–143 A new approach to Lipschitz continuity in state constrained optimal control1 A.L. Dontchev a, William W. Hager b;∗ a Mathematical Reviews, Ann Arbor, MI 48107, USA b Department of Mathematics, University of Florida, Gainesville, FL 32611, USA Received 22 January 1998; received in revised form 19 March 1998 Abstract For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discretization. Here we show that the theory developed for the Euler discretization can be used to derive the Lipschitz continuity of the optimal control. c 1998 Elsevier Science B.V. All rights reserved. Keywords: Optimal control; State constraints; Regularity; Discrete approximations 1. Introduction a ∈ Rn, and b ∈ Rk . Throughout, Lp denotes the usual Lebesque space of functions with integrable p-power, We consider the following linear-quadratic problem p ¿ 1, and H 1 is the space of absolutely continuous with state constraints: functions with derivatives in L2. We assume that Q Z is positive semideÿnite and R is positive deÿnite, and 1 1 minimize (x(t)TQx(t)+u(t)TRu(t)) dt (1) that there exists a pair (x; u) satisfying the constraints 2 0 (2)–(4). Under these conditions, there exists a unique ∗ ∗ subject to optimal solution (x ;u ) of problem (1). Let A denote the active constraint set deÿned by x˙(t)=Ax(t)+Bu(t) A(t)={j∈{1;2;:::;k}|K x∗(t)+b =0}; for a:e:t∈[0; 1];x(0) = a; (2) j j Kx(t)+b60 for all t ∈ [0; 1]; (3) where Kj is the jth row of the matrix K. We assume that the constraints satisfy the following regularity 2 1 u ∈ L ;x∈H; (4) condition introduced in [11]: where x(t) ∈ Rn;u(t)∈Rm, the matrices Q; R; A and Independence: A(0) = ∅ and there exists ÿ¿0 B have compatible dimensions, K is a k × n matrix, such that ∗ Corresponding author. E-mail: [email protected] .edu., http:== X www.math.u .edu= ˜hager. 1 This research was supported by the National Science vjKjB ¿ ÿ|v| (5) Foundation. j∈A(t) 0167-6911/98/$ – see front matter c 1998 Elsevier Science B.V. All rights reserved. PII: S0167-6911(98)00043-7 138 A.L. Dontchev, W.W. Hager / Systems & Control Letters 35 (1998) 137{143 for all t ∈ [0; 1] where A(t) 6= ∅ and for all v = This theorem is proved in [11] by applying a gen- {vj | j ∈ A(t)}. eral homotopy approach involving “compatible data”. Here we present another proof of this result using a dis- As shown in ([8], Eq. (43)), this regularity condi- crete approximation approach. Our proof goes along tion can be extended to the set of -active constraints; the following lines. We ÿrst show that the solution of a that is, there exists an ¿0 such that the independence discretized problem converges in an appropriate norm condition holds with the set A replaced by to the solution of the original problem. After show- ing the solution of the discrete problem is Lipschitz ∗ A(t)={j∈{1;2;:::;k}|Kjx (t)+bj¿−}: continuous in discrete time, with a Lipschitz constant independent of the mesh size, Theorem 1 is obtained The independence condition also implies that there ex- through a compactness argument. With the help of ists a feasible controlu ˜ such that the corresponding additional results from [9], the analysis in this paper trajectory x =˜xassociated with u =˜uin Eq. (2) satis- can be extended to handle control constraints, time- ÿes dependent matrices, and a weaker coercivity condition for the objective function. Kx˜(t)+b¡0 for all t ∈ [0; 1]: (6) In the case k = 1, corresponding to one state con- This fact follows from the more general results con- straint, a stronger result is obtained in [10]. Namely, tained in ([7], Lemma 3) or ([8], Lemma 3.6). under the independence condition, there exists an op- There are various forms available in the literature timal control which is a piecewise analytic function for the ÿrst-order optimality conditions (maximum of time. This follows from the more general result, principle) for the problem (1). For our purposes here, proved in [10], that in this case there are ÿnitely many it is convenient to use the following duality based re- points where the state constraint changes from active sult from Hager and Mitter [12]: If there is a feasible to nonactive. The generalization of this result to prob- pair (˜x; u˜) satisfying Eq. (6), then there exists a func- lems with multiple state constraints is an open prob- tion ∗ of bounded variation which is nondecreasing, lem. For recent works on regularity of solutions in right continuous, and with ∗(1) = 0, and a function optimal control, see [2–4, 13, 14]. ∗ ∈ H 1 which satisfy the following relations for al- Throughout this paper, c is a generic constant which is independent of time t and mesh size h, Var(f)isthe most every t ∈ [0; 1]: ÿ total variation of f on the interval [0; 1]; Var(f)| is ∗ −1 T T u (t)=R B (K (t)− (t)); (7) the total variation of f on the interval [ ; ÿ], Br(x) is the closed ball centered at x with radius r, and ˙(t)= −AT (t)+ATKT(t)−Qx∗(t); O(h) is an expression that is bounded by ch. (1)=0; (8) 2. The discrete problem Z 1 (Kx∗(t)+b)Td(t)=0: (9) 0 Let N be a natural number and deÿne h =1=N. The Euler approximation to problem (1) associated with Although u∗ only lies in L2 according to Eq. (4), it the grid ti = ih; i =0;1;:::;N, is the following: follows from Eq. (7) that there exists a member of the equivalence class associated with u∗ that has bounded NX−1 1 T T variation. Consequently, we assume henceforth that minimize x Qxi + u Rui (10) 2 i i u∗ is a particular element of the equivalence class i=0 that has bounded variation. In this paper, we establish subject to x0 = Ax + Bu ;x=a; Lipschitz continuity properties for the solution and the i i i 0 associated multipliers. Kxi +b60;i=0;1;:::;N −1; (11) Theorem 1. The multiplier ∗ is Lipschitz continu- 0 0 where xi is the ÿrst divided di erence, xi =(xi+1 − ous on [0; 1), while the equivalence classes associ- xi)=h. ∗ ated with x˙∗, u∗, and ˙ contain functions that are Utilizing the independence condition, we show be- Lipschitz continuous on [0; 1]. low that the problem (10) is feasible for h suciently A.L. Dontchev, W.W. Hager / Systems & Control Letters 35 (1998) 137{143 139 small. Since R is positive deÿnite and Q is semidef- Proof. By the state equation (2), we have inite, Eq. (10) has a unique solution (xh;uh). More- Z ti+1 over, by the Karush–Kuhn–Tucker conditions, there y(ti+1) − y(ti)= y˙(t)dt exist Lagrange multipliers ph and h such that the t Zi following optimality conditions hold: ti+1 = Ay(t)+Bv(t)dt 0 T T pi− = − Qxi − A pi − K i;pN−1=0; (12) ti −1 T =hAy(ti)+hBu(ti)+ri; ui = − R B pi; (13) where Kxi +b∈NRk (i); Z + ti+1 ri = A(y(t) − y(ti)) + B(v(t) − v(ti)) dt: for i =0;1;:::;N−1, where NRk () denotes the nor- + ti k mal cone to the positive orthant R+ at the point and 0 0 Combining this with Eq. (15) gives pi− is the backward divided di erence, pi− =(pi − pi−1)=h. h yi+1 = yi + hAyi + hB(v(ti) − v )+ri; The KKT multipliers ph and h are connected to the i multipliers and of the continuous problem through y0 =0; the following transformation: h where yi = y(ti) − yi . It follows that XN i = −h l; where N =0; and Xj−1 h l=i |yj| 6 c h|v(ti) − vi | + |ri| i=0 T = p + K : (14) i i i+1 N−1 Z X ti+1 0 6 c h|v(t ) − vh| + |y(t) − y(t )| We note that i = i. In terms of these transformed i i i i=0 ti multipliers, the ÿrst-order optimality conditions can ! be stated in the following way: There exist multipli- ers h and h associated with the solution (xh;uh)of + |v(t) − v(ti)| dt h h h h Eq. (10) such that (x ;u ; ; ) satisÿes the follow- ing conditions: NX−1 h ti+1 6 ch |v(ti) − v | + Var(y)| x0 = Ax + Bu ;x=a; (15) i ti i i i 0 i=0 ! 0 T T T i− = − A i − Qxi + A K i+1; N−1=0; (16) ti+1 + Var(v)|t T T T i Rui + B i − B K i+1 =0; (17) 0 NX−1 Kxi +b∈NRk (i): (18) h + 6 ch |v(ti) − vi | + ch(Var(y) + Var(v)): Our ÿrst lemma compares a continuous state trajectory i=0 to its discrete counterpart. This completes the proof. Lemma 1. Let v be a function of bounded variations As a consequence of Lemma 1, we have the exis- and let x = y be the solution of the state equation (2) tence of feasible points for (11).
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