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Second-order optimality conditions for singular extremals in optimal control problems with equality endpoint constraints. (English) Zbl 1373.49022

The problem is: minimize \(\varphi(x(T))\) among the trajectories of the initial value problem
\[ x'(t) = f(t, x(t), u(t)) \quad (S \leq t \leq T) \, , \quad x(S) = x_0 \, , \quad u(t) \in U(t) \] with equality constraint \(\psi(x(T)) = 0.\) Here \(x(t)\) is a \(n\)-dimensional vector, \(u(t)\) a \(m\)-dimensional vector, \(\psi(x)\) a \(p\)-vector function and \(\varphi(x)\) is real valued. A control-trajectory pair \((\bar x(t), \bar u(t))\) with \(\bar u(t)\) essentially bounded is an extremal of this problem if it satisfies Pontryagin’s maximum principle \[ H(t, \bar x(t), \bar u(t), p(t)) \geq H(t, \bar x(t), u, p(t)) \quad (S \leq t \leq T) \, , \quad u \in U(t) \eqno(1) \] where \(H(t, x, u, p) = p^Tf(t, x, u)\) and \(p(t)\) satisfies the final value problem \[ (p^T)'(t) = - H_x(t, \bar x(t), \bar u(t), p(t)) \quad (S \leq t \leq T) \, , \quad p^T(T) = - \lambda_0 \varphi'(\bar x(T)) - \lambda^T \psi'(\bar x(T)) \] with \((\lambda_0, \lambda) = (\lambda_0, \lambda_1, \dots \lambda_p) \neq 0.\) An extremal \((\bar x(t), \bar u(t))\) is singular in the sense of Pontryagin’s maximum principle if equality holds in (1) in some subinterval \(I\) of \([S, T],\) \[ H(t, \bar x(t), \bar u(t), p(t)) = H(t, \bar x(t), u, p(t)) \quad (t \in I) \, , \quad u \in V(t) \] where \(V(t) \subseteq U(t),\) \(V(t) \neq \{\bar u(t)\}.\) The authors compare this notion of singularity with various other definitions and derive second order optimality conditions for singular extremals.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N60 Regularity of solutions in optimal control
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