Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. CONTROL OPTIM. c  2008 Society for Industrial and Applied Mathematics Vol. 47, No. 1, pp. 535–552 NECESSARY CONDITIONS FOR CONSTRAINED PROBLEMS UNDER MANGASARIAN–FROMOWITZ CONDITIONS ∗ MARIA DO ROS ´ ARIO DE PINHO † AND JAVIER F. ROSENBLUETH ‡ Abstract. The focus of this paper is on first order necessary conditions for optimal control problems with mixed state-control equality and inequality constraints. We consider the case when the cost and dynamics are nonsmooth, and the constraints satisfy Mangasarian–Fromowitz-type as- sumptions that weaken the commonly used hypothesis that the Jacobian of the active constraints, with respect to the free variable, is of full rank. The results are formulated as unmaximized Hamil- tonian inclusion-type conditions involving not the customary product of partial subdifferentials but the joint subdifferential with respect to the state and control variables. Key words. optimal control, nonsmooth analysis, mixed constraints AMS subject classification. 49K15 DOI. 10.1137/060663623 1. Introduction. Consider the following optimal control problem with mixed constraints: (P) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Minimize l(x(0),x(1)) subject to ˙ x(t) = f (t, x(t),u(t),v(t)) a.e. in T, 0 = b(t, x(t),u(t),v(t)) a.e. in T, 0 ≥ g(t, x(t),u(t),v(t)) a.e. in T, v(t) ∈ V (t) a.e. in T, (x(0),x(1)) ∈ C, where T = [0, 1], and we are given functions l : R n × R n → R, (f,b,g): T × R n × R ku × R kv → R n × R m b × R mg , V (t) ⊂ R kv for all t ∈ T , and C ⊂ R n × R n . Let m := m b + m g , let k := k u + k v , and assume that k ≥ m. Usually one has m b ≥ 1 and m g ≥ 1, but we allow for m b =0 (no equality constraints) or m g = 0 (no inequality constraints). For (P), a process is a triple (x, u, v) comprising a function x ∈ W 1,1 (T ; R n ) and measurable functions u : T → R ku and v : T → R kv satisfying the constraints. Here W 1,1 (T ; R n ) denotes the space of absolutely continuous functions mapping T to R n . A process (¯ x, ¯ u, ¯ v) is a strong minimizer for (P) if there exists ǫ> 0 such that it minimizes the cost over all processes (x, u, v) satisfying |x(t) − ¯ x(t)|≤ ǫ for all t ∈ T , and it is a weak minimizer if, for some ǫ> 0, it minimizes the cost over all processes (x, u, v) satisfying |x(t) − ¯ x(t)|≤ ǫ for all t ∈ T and |u(t) − ¯ u(t)|≤ ǫ, |v(t) − ¯ v(t)|≤ ǫ a.e. in T. ∗ Received by the editors June 23, 2006; accepted for publication (in revised form) August 16, 2007; published electronically February 1, 2008. This work was supported by FEDER and FCT, Projecto POSC/EEA-SRI/61831/2004. http://www.siam.org/journals/sicon/47-1/66362.html † ISR and DEEC, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal (mrpinho@fe.up.pt). ‡ IIMAS–UNAM, Universidad Nacional Aut´onoma de M´ exico, Apartado Postal 20-726, M´ exico DF 01000, M´ exico (jfrl@servidor.unam.mx). 535