Abstract
In this paper we develop a new approach for optimal control problems with general jointly varying state endpoints (also called coupled endpoints). We present a new transformation of a nonlinear optimal control problem with jointly varying state endpoints and pointwise equality control constraints into an equivalent optimal control problem of the same type but with separately varying state endpoints in double dimension. Our new transformation preserves among other properties the controllability (normality) of the considered optimal control problems. At the same time it is well suited even for the calculus of variations problems with joint state endpoints, as well as for optimal control problems with free initial and/or final time. This work is motivated by the results on the second order Sturm–Liouville eigenvalue problems with joint endpoints by Dwyer and Zettl (1994) and by the sensitivity result for nonlinear optimal control problems with separated state endpoints by the authors (2018).
Highlights
In this paper we develop a new approach to optimal control problems with general jointly varying state endpoints, including the special case with periodic or antiperiodic endpoints
Necessary or sufficient optimality conditions for such optimal control problems are usually derived by a direct analysis of the problem, in particular when dealing with the nonnegativity or coercivity of the second variation, the Jacobi system and the Riccati differential equation, or the Hamilton–Jacobi theory, see e.g. [1, 2, 4, 5, 19, 26,27,28,29, 35, 36]
In the literature one can find an alternative approach based on transforming the problem with joint endpoints into an augmented problem with separated endpoints in double dimension
Summary
In this paper we develop a new approach to optimal control problems with general jointly varying state endpoints ( called coupled endpoints), including the special case with periodic or antiperiodic endpoints. In the literature one can find an alternative approach based on transforming the problem with joint endpoints into an augmented problem with separated endpoints in double dimension This method has been used in particular in the context of the Jacobi system (or the linear Hamiltonian system) and the corresponding Riccati equation, see e.g. In page 886 of [36], the transformation described above was suggested as a tool to extend to the joint endpoints case the results on optimality conditions established therein for the separated endpoints problem and involving the second variation phrased in terms of the associated linear Hamiltonian system and the corresponding Riccati differential equation. It is worth mentioning that the transformation developed in this paper is not restricted to the spaces of controls and states considered here, but it applies to any other spaces, e.g., to L∞ controls and absolutely continuous states
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