DCDS
Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval
Andrei V. Dmitruk Nikolai P. Osmolovskii
We study an optimal control problem with Volterra-type integral equation, considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the Euler--Lagrange equation and transversality conditions for the Lagrange problem in the classical calculus of variations with ordinary differential equations.
keywords: transversality conditions change of time variable. Euler--Lagrange equation adjoint equation Volterra-type equation extended weak minimum
DCDS
Quadratic order conditions for an extended weak minimum in optimal control problems with intermediate and mixed constraints
Andrei V. Dmitruk Alexander M. Kaganovich
We consider a general optimal control problem with intermediate and mixed constraints. Using a natural transformation (replication of the state and control variables), this problem is reduced to a standard optimal control problem with mixed constraints, which makes it possible to obtain quadratic order conditions for an "extended" weak minimum. The conditions obtained are applied to the problem of light refraction.
keywords: Intermediate constraints weak minimum mixed constraints replication of state and control variables. Euler--Lagrange equation second variation
NACO
Quadratic order conditions for bang-singular extremals
M. Soledad Aronna J. Frédéric Bonnans Andrei V. Dmitruk Pablo A. Lotito
This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.
keywords: second order condition singular arc control constraint bang-singular solution. Optimal control
MCRF
Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval
Andrei V. Dmitruk Nikolai P. Osmolovski

We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary differential equations.

keywords: Volterra-type equation extended weak minimum local maximum principle state constraints mixed state-control constraints adjoint equation transversality conditions change of time variable linear–positive independence

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