An optimum settling problem for time lag systesm

Control of Functional Differential Equations With Function Space Boundary Conditions

“…As can be seen with relatively straightforward arguments, the conditions obtained in [6] and [22] are essentially equivalent (with one exception) whenever the problem is such that the results of both papers are applicable.…”

“…Finally, Jacobs and Kao [22] (see also [6]) show that the usual weak compactness arguments in L p can be used to obtain existence theorems for problems of the type being considered with linear retarded system equations and Lagrange payoffs with quadratic integrands. Gamkrelidze [l6].…”

“…However, the authors do discuss a number of solved examples which demonstrates that the class of problems for which the conditions are necessary and sufficient, and for which they offer nontrivial conditions, is non- Jacobs and Kao [ 22] also consider the Lagrange problem above but for retarded systems with delays in the controls x(t) = f (x(t),x(t-h),u(t), "u(t-t),t). Using the Lagrange Multiplier Rule [ 28] in ¥" for an unconstrained class of controls ^= L ?…”

“…Using a class of unconstrained controls ^ = Lp([ t-., t..],R ) they eniploy attainable sets arguments (see [27]) in Wp ' ([ t..-h,t_]),R ) to de--rive-essentially the conditions in Theorem ^.1-with -normality (and-hence sufficiency) assured. The assumptions under which normality is guaranteed (controllability type assumptions on the system and the relatively weak hypothesis on BI p(t) = rank B(t) is constant on [t, -h,t..]) are muchless restrictive than those in [22], As is pointed out in [7], these assumptions are weak enough to include the general n order scalar neutral equation with scalar control in the class of systems to which the results are applicable.…”

On the necessary conditions for optimal control of retarded systems

“…The papers listed above exhibit the kinds of difficulties which are met: either the nontriviality of multipliers cannot be guaranteed (see, e.g., [5]) or pointwise control constraints cannot be taken into account [3], [14], or consideration must be restricted to systems whose dynamics are linear in the control variable, admissible control functions being required to lie in some £~'P-class for 1 < p < (see [10] and [15]). …”