Quadratic conditions for nonsingular extremals in optimal control (a theoretical treatment)
The subject of this paper is an optimal control problem dealing with the dynamic controlled object given by: \[ \dot{x}=f(x, u, t), \] \[ g(x, u, t)=0,\quad h(x, u, t) \leq 0, \] \[ (x(t_{0}), x(t_{1})) \in P,\qquad (x, u, t) \in Q \] where \(P\) and \(Q\) are open sets, \(x, u, f, g, h\) are the vector functions fulfilling the inequality \(d(g) \leq d(u)\). There is considered the following control optimal problem on a fixed time interval \([t_{0}, t_{1}]\) \[ \text{minimum }\{ J(x, u)=(F_0(x(t_{0}), x(t_{1})) \} \] subjected to the constraints \[ F(x(t_{0}), x(t_{1})) \leq 0, \quad G(x(t_{0}), x(t_{1})) = 0 \] where \(F\) and \(G\) are also vector functions fulfilling the same inequality as above, i.e., \(d(g) \leq d(u)\). Necessary conditions and quadratic sufficient conditions have been stated for different types of minima in the canonical optimal control problem. Jacobi-type conditions have been used in an examination of the definiteness of the quadratic form. In the time-optimal control problem for a linear systems the general quadratic conditions have been reduced to a quite simple form in a finite-dimensional space. The work establishes a survey of results of investigations published previously by the same author.
necessary and sufficient conditions for optimal solution
scientific article; zbMATH DE number 1206245