Abstract
We derive necessary optimality conditions for the time of crisis problem under a more general hypothesis than the usual one encountered in the hybrid setting, which requires that any optimal solution should cross the boundary of the constraint set transversely. Doing so, we apply the Pontryagin Maximum Principle to a sequence of regular optimal control problems whose integral cost approximates the time of crisis. Optimality conditions are derived by passing to the limit in the Hamiltonian system (without the use of the hybrid maximum principle). This convergence result essentially relies on the boundedness of the sequence of adjoint vectors in L∞. Our main contribution is to relate this property to the boundedness in L1 of a suitable sequence which allows to avoid the use of the transverse hypothesis on optimal paths. An example with non-transverse trajectories for which necessary conditions are derived highlights the use of this new condition.
Highlights
This paper proposes a novel approach to derive optimality conditions for the so-called time of crisis problem [7] as well as sufficient conditions ensuring the well-posedness of this method
The time of crisis arises in the context of viability theory [2, 3], see, e.g., a case study in ecology in [9], and more generally whenever one is unable to maintain a controlled dynamics within a prescribed constraint set over a time window
We have developed an approach based on a sequence of approximate optimal control problems
Summary
This paper proposes a novel approach to derive optimality conditions for the so-called time of crisis problem [7] as well as (new) sufficient conditions ensuring the well-posedness of this method. As in [17], this hypothesis is crucial for the derivation of necessary conditions (in particular, for an accurate definition of the jump of the covector at a crossing time) Thanks to this hypothesis, it is shown in [7, 8] that extremals of a regularized optimal control problem converge, up to a sub-sequence, to an extremal of the time of crisis problem. The last section is devoted to a complete study of an illustrative example that highlights the convergence of the sequence of adjoint vectors to a discontinuous covector having a jump at each crossing time This phenomenon is depicted in a transverse situation, and when a trajectory hits the boundary of the constrained set tangentially (and leaves the set non-tangentially)
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