OPTIMAL CONTROL OF NONSMOOTH SYSTEM GOVERNED BY QUASI-LINEAR ELLIPTIC EQUATIONS GONG

In this paper, we discuss a class of optimal control problems of nonsmooth systems 
governed by quasi-linear elliptic partial differential equations, give the existence of the problem. Through 
the smoothness and the approximation of the original problem, we get the necessary condition, which can 
be considered as the Euler-Lagrange condition under quasi-linear case.

Consider the problem (ocp), Casas and Fernandez [5] studied the special case when a/(1 < < n) are continuously differentiable and J(u -lY Yd dx / - got the necessary condition, Barbu [4] studied the nonsmooth linear systems and got the Euler-Lagrange condition, etc But for the nonsmooth quasi-linear systems, there are no conclusions yet In this paper, we discuss this problem, give the existence and get the necessary condition of it through the smoothness of a(x), G(y) and the approximation of (ocp).Also this way can be used to study the boundary control problem 2. TIlE EXISTENCE OF SOLUTION OF (ocp) To get the existence of (ocp), we need to study the Dirichlet problem (1.1) first From Ladyzhenskaya [2], we get.LEMMA 2.1.Let u E L2(/) satisfy llUllL2(n) < M, and the Dirichlet problem (1.1) satisfy (1.2), (1 3), (1 4) Then" 1) There exists a unique y W0 'v fq L2(f) be a solution of(1 1) and a constant C > 0 depending only on M, w, r/, c, such that 2) Vu, L2(f2)(rn N), (1.1) has solutions Yu Y, assume: u,-u weakly in Lg(f) as mc.
Then, yy strongly in Y.
In this paper we denote Y to be W0 (l)) fq L2 (I2) From the second part ofLemma 2.1 we can define an operator O L2 (12) Y as () =.
Applying Lemma 2. we can get the existence theorem of (ocp).THEOREM 2.2.There exists at least one solution ofthe optimal control problem (ocp), we denote it [, ].PROOF.Suppose that {u,} is a minimizing sequence of (ocp), because ofthe coercivity of J(y, u), we get the boundness of Ilu, llL2(n), then there exists a subsequence u, (denoted the same way) and L2 (f2), such that u---,, weakly in L2(f) as i-,oo.
From Lemma 2.1 we get y, y (denoted ) strongly in Y, and the lower semicontinuity of J(y, u) shows J(, ) lim inf J (y, u) _< J (y, u), V u E Lg (f/) i.e [, ] is a solution of (ocp).

NECESSARY CONDITION OF THE SOLUTION OF (ocp)
Before studying the necessary condition of (ocp), we need the definition of the Generalized Gradient and the Yosida regularization of the Lipschitz function (Tiba D. [ 10]).DEFINITION 3.1.If F(x) is a local Lipschitz function, its Generalized Gradient denoted DF(x) is the convex hull of the set of cluster points for the sequences grad(x + h,), where h, 0 are chosen such that grad(x + h,) exist, i.e.DF(x) conv{w Rr', :::1 h, O, 3 grad F(x + h,) w}.where p(z) is a C (f2) function satisfy Obviously, a'(z) is a C function and a(z) a,(z) uniformly as e 0, an elementary calculation shows that.
Compared with the original operator O, we have: Then, y'y strongly in H0 (f), as -0.
Ne we prove the threm tough 3 steps.
The sequence {} is bound in H (). yX_y} , is bounded in H (f2). STEP 2. We denote z x we will get a subsequence and prove the boundness of rnz" in From (3.7), we get a subsequence z (denote in the same way) such that z x r, weakly in H (f).
Next we will prove h d: (y,)r,, a.e. on f Since and Z A m A Z z r, weakly in L2 (f) From Lemma 2 we get y, y, in L2(f/).By Egorov Theorem we know Va > 0,3f2o C f, such that rn(f f2o) < a, and y, yx, uniformly in L2(f2).