MAXIMUM PRINCIPLES FOR OPTIMAL CONTROL PROBLEMS WITH DIFFERENTIAL INCLUSIONS

There are three different forms of adjoint inclusions that appear in the most advanced necessary optimality conditions for optimal control problems involving differential inclusions: EulerLagrange inclusion (with partial convexification) [A. D. Ioffe, J. Optim. Theory Appl. , 182 (2019), pp. 285--309], fully convexified Hamiltonian inclusion [F. H. Clarke, Mem. Amer. Math. Soc. , 173 (2005), 816], and partially convexified Hamiltonian inclusion [P. D. Loewen and R. T. Rockafellar, SIAM J. Control Optim. , 34 (1996), pp. 1496-1511], [A. D. Ioffe, Trans. Amer. Math. Soc. , 349 (1997), pp. 2871-2900], [R. B. Vinter, SIAM J. Control Optim. , 52 (2014), pp. 1237--1250] (for convex-valued differential inclusions in the first two references). This paper addresses all three types of necessary conditions for problems with (in general) nonconvex-valued differential inclusions. The first of the two main theorems, with the Euler-Lagrange inclusion, is equivalent to the main result of [A. D. Ioffe, J. Optim. Theory Appl. , 182 (2019), pp. 285--309] but proved in a substantially different and much more direct way. The second theorem contains conditions that guarantee necessity of both types of Hamiltonian conditions. It seems to be the first result of such a sort that covers differential inclusions with possibly unbounded values and contains the most recent results of [F. H. Clarke, Mem. Amer. Math. Soc. , 173 (2005), 816] and [R. B. Vinter, SIAM J. Control Optim. , 52 (2014), pp. 1237--1250] as particular cases. And again, the proof of the theorem is based on a substantially different approach.