Lipschitz Controls For A Class Of Second Order Ordinary Differential Equations

The main interest of this paper is to study a Lipschitz control problem associated to a second order ordinary differential equations with pointwise state constraints. More precisely, we shall investigate the following control problem:The state equation iswhere f(x, y) is continuous differentiable and satisfies the following conditions:In addition, the state constraints areWe shall consider the admissiable control setHere, a, b, l, k∈R satisfying 0ad such that the corresponding state minimizing this cost functional.In our problems, the control is realized via the function of the state. There is a few of papers dealing with problems considered here. The paper by R. Kluge [1] and the very recent papers by V. Barbu, Kunisch and W. Ring ([2], [3], [4]) and A. Rosch [5] are devoted to similar problems for partial differential equations. M. Goebel and D. Oestreich [6] 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 equation in the form y"(x) +σ(y(x)) = 0, and obtained some results. In [8], M. Akkouchi, A. Bounabat and M. Goebel extended the results of M. Goebel [7] to the case of the state equation in the formHere, we consider the state equation is in the formWe first show the existence and uniqueness of the solution of the state equation and the existence of the optimal controls. In addition, we obtain necessary optimality conditions for P .For a nonsmooth optimal control functionσ, we introduce the set given byand define two new functions (?) by settingwe introduce the new setΦ0(σ) given by Now, let us denote byσ0 any given Lipschitz continuous extension of the (possibly nonsmooth) controlσ0: [-a, a]→R to the whole R , and letσ₀⁰(x, (?)) designates Clarke's directional derivative ofσ0 at x in the direction (?) for all x, (?)∈R. Then with these notations we have the following result.Theorem. Letσ0∈∑_ad be an optimal control to P and y0∈C([0,1]) the related optimal state. Then for allσ0∈∑_ad and all (?)∈Φ0(σ) it holds:For a smooth optimal control function, we obtain the following necessary op-timality condition:Theorem. If is an optimal control to the problem P, y0(x) the related optimal state, then it holdswhere z0∈C([0,1]) is the unique solution to the linear boundary value problem To this end, we introduce the closed subset∑_ad¹ of C¹([-a, a]) given by By means of Ekeland's variational principle, we derive the following subopti-mality condition:Theorem. Letσ0 be an optimal control to the problem P. Then we have:(ii) For anyα> 0, there exists an elementσ_α∈∑_ad¹, such that with the associated state yα∈C([0,1]) it holds:where is the state related toσ, and z_α∈C([0,1]) denotes the unique solution to the linear boundary value problem...