Problems of the positional control of systems in a class of generalized equations with delay (Q2755742)

The paper deals with a positional control problem for systems with control delays NEWLINE\[NEWLINE \dot x(t)=f_1(t,x(t),u(t),u(t-\tau))+f_2(t,x(t),v(t)), NEWLINE\]NEWLINE where \(t\in [t_0,\theta],\) \(\tau>0,\) \(x\in \mathbb{R}^n,\) \(u\in P,\) \(v\in Q;\) \(P\) and \(Q\) are compacts. In the work by \textit{Yu. S. Osipov} and \textit{V. G. Pimenov} [``On positional control under delay in control parameters'', Prikl. Mat. Mekh. 45, No. 4, 223-229 (1981; Zbl 0498.90099)] there were obtained solutions of the approach-evasion problems with a functional set in the space of triples \(\{t, x, u_t(\cdot)\}\), \(u_t(\cdot)= \{u(t+s), -\tau\leq s<0 \}\). Motions of such systems turn out to be non-compact in the position space because of the delay in control parameters. The author introduces a notion of generalized controls \(\mu(t)\) which are the weak measurable function to the set \(rpm(P\times P)\) with the additional consistence condition depending on the delay and an initial function (according to the scheme of the work \textit{V. G. Pimenov} [``Concept of the generalized controls for functional-differential systems'', Differ. Uravn. 31, No. 6, 980-989 (1995; Zbl 0862.49009)]). This allows to introduce not only approximation motion of systems but also their limit motions and to avoid the procedure of extreme shift to the sequence of sets. In the paper the problems of approach-evasion with the set in the space of triples \(\{t, x, \mu_t(\cdot)\}\) are formalized. The solving constructions and alternative theorem are derived. The conditions are obtained under which the control problems for classical and generalized controls are equivalent.