Degeneracy and generalized solutions of optimal control problems∗)

Degenerate optimal control problems are often met in applications and bring serious diffi-culties for general methods. They were studied systematically beginning from the 1960-s [1-4], and defined (generalizing the classical notion of degeneracy) as those which have some latent passive differential constraints. We propose special methods based, in essence, on detection and elimination of passive constraints. The core of this theory is a certain transformation of systems x ̇ = f(t, x, u), t ∈ T = [tI, tF], x ∈ Rn, u ∈ U(t, x) ⊂ Rp (1) with unbounded velocity set V(t, x) = f(t, x,U(t, x)). Namely, we introduce an auxiliary system which describes the behavior of the original system for large velocities (called limit system): dx dτ = l ∈ K(t, x). (2) Here t is a parameter and K is the union of all limits l = lim vq|vq|−1 as |vq | → ∞, {vq} ⊂ V