Maximum Principles for Optimal Control Problems with Differential Inclusions

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 271-296, February 2024.
Abstract. There are three different forms of adjoint inclusions that appear in the most advanced necessary optimality conditions for optimal control problems involving differential inclusions: Euler–Lagrange 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.