Shape sensitivity analysis via min max differentiability. (English) Zbl 0654.49010

Delfour, M. C.; Zolésio, J.-P.

SIAM J. Control Optimization 26, No. 4, 834-862 (1988).

The authors present new theorems on the differentiability of the minimax: \(g(t)=\) \(\{\) min[max G(t,x,y): \(y\in B]:\) \(x\in A\}\) of a functional G(t,x,y) with respect to a real parameter \(t\geq 0\) for some fixed subsets \(A\subset X\) and \(B\subset Y\) of two topological spaces X and Y. At first they consider a special and useful case when G(t,x,y) is a convex-concave functional with unique saddle point \((x\) \(*_ t,y\) \(*_ t)\) in \(A\times B\) for each t. Then they show that under some reasonable hypotheses \(dg(t)/dt=\partial_ tG(t,x\) \(*_ t,y\) \(*_ t)\). In control theory this is a natural tool in the computation of the directional derivative of the cost function.
In the main result, G(t,x,y) is no longer assumed to be a convex-concave functional with saddle point. The authors prove that under appropriate hypotheses the function g is differentiable at \(t=0\) from the right. This theorem has many applications. For example, the first is a control problem with a nondifferentiable observation which depends on the state which is the solution of an elliptic equation which itself depends on the control function u. The second is the shape analysis problem.

Reviewer: M.Todorov

MSC:

49K40	Sensitivity, stability, well-posedness
49J52	Nonsmooth analysis
49J35	Existence of solutions for minimax problems
49J40	Variational inequalities
35R35	Free boundary problems for PDEs
93C20	Control/observation systems governed by partial differential equations

Keywords:

minimax; control problem with a nondifferentiable observation; shape analysis