Weak maximum principle and accessory problem for control problems on time scales. (English) Zbl 1157.49030

Summary: We derive the first and second variations for a nonlinear time scale optimal control problem with control and state-endpoints equality constraints. Using the first variation, a first order necessary condition for weak local optimality is obtained under the form of a weak maximum principle generalizing the Dubois-Reymond Lemma to the optimal control setting and time scales. A second order necessary condition in terms of the accessory problem is derived by using the nonnegativity of the second variation at all admissible directions. The control problem is studied under a controllability assumption, and with or without the shift in the state variable. These two forms of the problem are shown to be equivalent.


49K15 Optimality conditions for problems involving ordinary differential equations
39A12 Discrete version of topics in analysis
34K35 Control problems for functional-differential equations
49N25 Impulsive optimal control problems
93B05 Controllability
93C15 Control/observation systems governed by ordinary differential equations
93C55 Discrete-time control/observation systems
Full Text: DOI


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