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Necessary conditions for a minimum for nonregular problems in Banach spaces. A maximum principle for abnormal optimal control problems. (English. Russian original) Zbl 0709.49007

Proc. Steklov Inst. Math. 185, 1-32 (1990); translation from Tr. Mat. Inst. Steklova 185, 3-29 (1988).
The author considers the following control problem with fixed time: \[ (1)\quad F(x(\cdot),u(\cdot))=\int^{t_ 1}_{t_ 0}f^ 0(t,x(t),u(t))dt\quad \to \quad \inf, \]
\[ (2)\quad \dot x=f(t,x,u), \]
\[ (3)\quad u\in U=\{u(\cdot)\in L^ r_{\infty}([t_ 0,t_ 1])| \quad u(t)\in V\quad a.e.\}, \]
\[ (4)\quad g_ 0(x(t_ 0))=0,\quad g_ 1(x(t_ 1))=0, \] where the function \(f^ 0\) and the mappings f, \(g_ 0\) and \(g_ 1\) are twice continuously differentiable with respect to x.
A pair \((x(\cdot),u(\cdot))\in W^ n_{1,1}([t_ 0,t_ 1])\times U\) is called an abnormal or a nonregular regime of constraints (2)-(4) if it satisfies (2)-(4) and, moreover, there exist vectors \(\ell_ 0\in {\mathbb{R}}^{k_ 0}\) and \(\ell_ 1\in {\mathbb{R}}^{k_ 1}\), not simultaneously zero, and a vector-valued function \(\psi\) (\(\cdot): [t_ 0,t_ 1]\to {\mathbb{R}}^ n\) such that \[ {\dot \psi}=-f^ T_ x(t,x(t),u(t))\psi,\quad \psi (t_ 0)=g^ T_{0x}(x(t_ 0))\ell_ 0, \]
\[ \psi (t_ 1)=-g^ T_{1x}(x(t_ 1))\ell_ 1,\quad (\psi (t),f(t,x(t),u(t))\equiv ({\dot \psi}(t),f(t,x(t),u)) \] for all \(u\in V\) and almost all \(t\in [t_ 0,t_ 1]\). Problem (1)-(4) is called an abnormal or a nonregular optimal control problem if an optimal process \(x_*(\cdot),u_*(\cdot)\) belongs to the set of nonregular regimes of the constraints.
The main aim of the author is to determine first order necessary conditions in the form of a maximum principle for the abnormal case. The proof is based on a general scheme for smoothly convex abnormal problems in a Banach space.
Reviewer: L.Mikołajczyk

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
49M29 Numerical methods involving duality
49K27 Optimality conditions for problems in abstract spaces
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