Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Abstract

Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. problems with a constant state delay. In [3], he gave similar results for control problems with pure control delays. Halany [4] proves a maximum principle for optimal control problems with multiple constant delays in state and control variables that, however, are chosen to be equal for state and control. Similar results were obtained by Ray and Soliman [5]. Guinn [6] sketches a simple method for obtaining necessary conditions for control problems with a constant delay in the state variable. He suggests to augment the delayed control problem that yields a higher-dimensional undelayed control problem to which the standard maximum principle is applicable. Banks [7] derives a maximum principle for control systems with a time-dependent delay in the state variable. Delays in the control are admitted for systems linear in the control variable. Colonius and Hinrichsen [8] provide a unified approach to control problems with delays in the state variable by applying the theory of necessary conditions for optimization problems in function spaces. All articles mentioned so far do not consider general control or state inequality constraints.Angell and Kirsch [9] treat functional differential equations with function-space state inequality constraints. However, they do not discuss the regularity of the multiplier associated with the state constraint and do not provide a numerical example with a pure state space constraint. To our knowledge, optimal control problems with constant delays in state and control variables and mixed control-state inequality constraints have not yet been considered in the literature. The first goal in this paper is to derive a Pontryagin-type minimum (maximum) principle for this class of delayed control problems. Concerning the development of numerical methods and the numerical treatment of practical examples, our impression is that this topic has not yet been adequately addressed in the literature. Bader [10] uses collocation methods to solve the boundary value problem for the retarded state variable and advanced adjoint variable. He successfully solves several academic examples, but his method does not give accurate results for the more difficult CSTR...

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