zbMATH — the first resource for mathematics

On variational analysis of differential inclusions. (English) Zbl 0761.49003
Optimization and nonlinear analysis, Proc. Binatl. Workshop, Haifa/Israel 1990, Pitman Res. Notes Math. Ser. 244, 199-213 (1992).
[For the entire collection see Zbl 0745.00050.]
The paper is devoted to a Mayer optimization problem for convex-valued differential inclusions on a varying time interval with nonsmooth endpoint constraints. We develop a finite difference method for studying this problem and prove the uniform convergence of optimal solutions for discrete approximations to the given optimal trajectory for the initial differential inclusion. Then we provide a variational analysis of discrete approximation problems employing suitably generalized differentiation constructions for nonsmooth mappings and sets. The main attention is paid to the so-called (nonconvex) coderivation of multifunctions which appears to be an appropriate construction for describing the adjoint difference and differential inclusions. Using these tools, we prove necessary optimality and controllability conditions for the initial problem in a generalized Euler-Lagrange form. These conditions contain the maximum principle, refined transversality inclusions with a new relation for the optimal time interval, and a differential inclusion for the adjoint arc. The latter adjoint inclusion requires less convexification in comparison with the Euler-Lagrange inclusion in Clarke’s form.

49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
49J52 Nonsmooth analysis