Smooth And Nonsmooth Lipschitz Controls For A Class Of Vector Differential Equations

In this paper, our main interest is to study a Lipschitz control problem associated to a nonlinear second order vector differential equations with point-wise stale constraints. More precisely, we shall investigate the following control problem:The stale equation is:where p C ([0. 1].Rn n) satisfies that p is symmetric and positive delinite. f(t,r) is continuous differentiabie and satisfies the following conditions:(H1) f(t,0) = 0. f (0,1);In addition, the stale constraints areWe shall consider the admissiable control setFor a given vector-valued function h in the Banaeh space C([0,1].Rn).we introduce the cost functionalwhere denotes the standard inner product in Rn. denotes the Euclidean norm of vector or the compatible matrix norm, a,b,k,l are fixed positive real numbers, and G = {x Rn. |x| a}. We want to find some ad such that the corresponding state minimizes this cost functional.In our problems, the control is realized via the funetion of the stale. There is a lew of papers dealing with problems considered here. The paper by R. Kluge[11]. and the very recent papers by Y. Barbu. K. Kunisch and W. Ring[2]|3][4]. and A. R sch[14] are devoted to similar problems for partial differential equations. M. Goebel and D. Oestreich[9] deal with an optimal control problem of such a type, however for a nonlinear singular integral equation. Recently. M. Goebel[7] considered the simple case of the state equal ion in the form y(x) + (ii(.r)) - 0. and obtained some results. In [1]. M. Akkouchi. A. Bounabat and M. Goebel extended the results of M. Goebel[7] to the case of the state equation in the form (p(t)x'(t))' + q(t)x(t) + (x(t) = .- How-ever. they all only considered the scalar state equation. Here, we extend the results of scalar differential equations to vector differential equations, and fur-thermore consider the more general case, i.e., the state equation is in the form This paper consists of two parts.In the first part (chapter 2). we consider the special optimal control problem (P0) where f(t,x) = q(t)r,i.e. the state equation is in the formWe make the following, assumption:(H3) Let A(t) be the smallest eigenvalue function of p(f), (t) the greatest eigenvalue function of q(t). We setsuch thatUnder the above assumption. we first show the existence and uniqueness of the solution of the state equation and the existence of the optimal controls. In addition, we obtain two necessary optimally conditions depending on their smoothness properties.For a smooth optimal control function, we obtain the following necessary optimality condition:Theorem. If C1(G) is an optimal control to the problem(P0). x0(t) C([0, 1],Rn)the related oi)t.iiiia] state, then it holdswhere z0(t) f'([0. 1],Rn) is the unique solution to the linear BVTHere, D denotes the linearized operator of a vector-valued function.For a nonsmooth optimal control function, by means of Ekelaud's variationsl ])rinciple we deri\-e a snl)optimality condiiion:Theorem. Let 0 he an optimal control to the problem (P0). Then we have(ii) For any (I there exists an element with the state . such thatwhere is the state related to . denotes the nuique solution to the BVPIn the second part (chapter 3), we consider the more general optimal control problem (P), i.e., the state equation is in the form (1), where f(t,x) satisfies conditions (HI) and (H2). Moreover, we shall make the following assumption:(H4) Let (t) be the smallest eigenvalue function of p(/). We setsuch thatUnder these assumptions, we firstly prove the existence and uniqueness of the solution of the state equation and the existence of the optimal controls. Then, by using the linearization technique, along the line of chapter 2, we obtain two necessary optimality conditions depending on their smoothness properties which are similar to the results of chapter 2.For a smooth optimal control function, we have the following necessary optimality condition:Theorem. If 0 ad C1(G) is an optimal control to the proble...