Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost

By Emmanuel Trélat. 2000

84 citations recorded by the source corpus.

AbstractLet $$T >0$$ be fixed. We consider the optimal control problem for analytic affine systems: $$\dot x = f_0 (x) + \sum\limits_{i = 1}^m {u_i f_i (x)} $$ , with a cost of the form: $$C(u) = \int\limits_0^T {\;\sum\limits_{i = 1}^m {\;u_i^2 (t)\;dt} } $$ . For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function S is subanalytic. Second, we prove that if there exists an abnormal minimizer of corank 1, then the set of endpoints of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/43adf8196e9ea1fe90ae020276e3efba7e2b6bf7

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Morse-Sard type results in sub-Riemannian geometry (2005, background): “As a consequence, since the end-point mapping Ex0 is continuous for the weak L2-topology (see [ 9 ] for a proof), we deduce,”
Semiconcavity results for optimal control problems admitting no singular minimizing controls (2008, background): “Recalling that the map (t, u(·)) → dE0 (u(·)) is continuous in the weak L1-topology (see [24]), in the limit as k → ∞ we obtain that there exist p̄ ∈ R with |p̄| = 1 and a pair (t, u(·)) ∈ K, with u ∈ L2([0, t];Rm), such that 〈p̄, dE0 (u(·))(v(·))〉 = 0 for every v(·) ∈ L2([0, t];Rm).”