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IMA Journal of Mathematical Control and Information 1999 16(4):335-351; doi:10.1093/imamci/16.4.335
© 1999 by Institute of Mathematics and its Applications
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A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control

F. RAMPAZZO {dagger} and R. B. VINTER {ddagger}

Dipartimento di Matematica Pura e Applicata, Universita di Padova 35131 Padova, Italy
Centre for Process Systems Engineering and Department of Electrical and Electronic Engineering, Exhibition Road London SW7 2BT, UK

Simple directly verifiable conditions are derived under which there exists a state trajectory satisfying a specified state constraint. The conclusions differ from the kind of information provided by viability and invariance-type theorems, insofar as an estimate is provided of the distance (in the supremum norm) of the state trajectory from a specified state trajectory, in terms of the degree to which the specified state trajectory violates the state constraint. The constructions involved in the existence proof are related to ones previously employed by Soner to establish continuity properties of a value function arising in infinite-horizon state-constrained optimal control, but the accompanying analysis contains refinements to ensure a sub-Lipschitz property of the value function considered here. It is expected that this existence result will have a number of implications for systems theory and optimal control. Here we show how it leads to a non-degenerate maximum principle for state-constrained optimal-control problems, in situations where the standard necessary conditions give no useful information about minimizers.

{dagger} Email: rampazzo{at}pdmat1.unipd.it

{ddagger} Email: r.vinter{at}ic.ac.uk


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