Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints
This article deals with necessary quadratic (second-order) conditions of extremum in optimal control problems with control appearing nonlinearly. The main difference from other publications is that here the necessary conditions take into account discontinuities of the first kind of the optimal control. The control problems under investigation are governed by ordinary differential equations with initial-final state constraints and mixed state-control constraints (of equality and inequality type). All the results included in this long article are based on the author's doctoral thesis, defended in 1988 at the Moscow Civil Engineering Institute. The author's necessary conditions are expressed in terms of positive semi-definiteness of the quadratic form on the so-called critical cone. In the case of discontinuous optimal control, the definitions of the quadratic form and critical cone are more complicated than in the continuous case. The natural strengthening of these necessary conditions turns them into sufficient conditions. The paper contains five chapters. In Chapter 1 the necessary quadratic conditions are derived under convenient assumptions. All data are assumed to be twice continuously differentiable and the gradients with respect to the control of active mixed constraints are assumed to be linearly independent. The reference control is assumed to be piecewise continuous on the corresponding time interval. The first- and second-order necessary conditions for optimality are formulated for the case of discontinuous optimal control. Chapter 2 is devoted to the abstract theory of higher-order conditions. In Chapters 3 and 4 the author provides proofs of the results presented in Chapter 1. Chapter 5 is devoted to some examples illustrating the applicability of the second-order necessary optimality conditions. The article is self-contained. It is definitely useful for advanced students and researchers interested in optimal control problems.
necessary second-order optimality conditions
discontinuous (piecewise continuous) control
scientific article; zbMATH DE number 6110970
scientific article