On guarantee optimization in control problem with finite set of disturbances

In this paper, we deal with a control problem under conditions of disturbances, which is stated as a problem of optimization of the guaranteed result. Compared to the classical formulation of such problems, we assume that the set of admissible disturbances is finite and consists of piecewise continuous functions. In connection with this additional functional constraint on the disturbance, we introduce an appropriate class of non-anticipative control strategies and consider the corresponding value of the optimal guaranteed result. Under a technical assumption concerning a property of distinguishability of the admissible disturbances, we prove that this result can be achieved by using control strategies with full memory. As a consequence, we establish unimprovability of the class of full-memory strategies. A key element of the proof is a procedure of recovering the disturbance acting in the system, which allows us to associate every non-anticipative strategy with a full-memory strategy providing a close guaranteed result. The paper concludes with an illustrative example.

Set-Invariance-based Interpretations for the L1 Performance of Nonlinear Systems

This paper is concerend with tackling the L 1 performance analysis problem of continuous and piecewise continuous nonlinear systems with non-unique solutions by using the involved arguments of set-invariance principles. More precisely, this paper derives a sufficient condition for the L 1 performance of continuous nonlinear systems in terms of the invariant set. However, because this sufficient condition intrinsically involves analytical representations of solutions of the differential equations corresponding to the nonlinear systems, this paper also establishes another sufficient condition for the L 1 performance by introducing the so-called extended invariance domain, in which it is not required to directly solving the nonlinear differential equations. These arguments associated with the L 1 performance analysis is further extended to the case of piecewise continuous nonlinear systems, and we obtain parallel results based on the set-invariance principles used for the continuous nonolinear systems. Finally, numerical examples are provided to demonstrate the effectiveness as well as the applicability of the overall results derived in this paper.

Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx whose coefficients constitute a convex slowly varying sequence b. The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g(b,x) from the main term of its asymptotics bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(x)/x in some neighborhood of zero admits a two-sided estimate in terms of the function σ(b,x) with absolute constants independent of b. Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g(b,x) on the class of convex slowly varying sequences in the regular case is obtained.

Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order $1<\alpha <2$ 1 < α < 2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval $J=[0,\infty )$ J = [ 0 , ∞ ) . We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

Controlled Process of Crystallization in Weld Pool

In this report the problem of control of solidification crack formation in welded plates is considered. In this problem the welding source is determined in dependence on the preset configuration and curvature of the rear part weld pool. A double ellipsoid model of weld pool with preset semi-axes may be used for the first approximation of preset weld pool configuration. It is an inverse problem which may be more efficiently solved as optimal control problem. The Function of welding source as a controlling function obtained in the result of solution is determined in a class of piecewise continuous functions which is more common class and the continuous-smooth functions are special partial case of common class. Recent methods of optimal control which use for solution of optimal control problems require to present the controlling functions in class of piecewise constant functions. Laser influence, electron beam, plasma, arc and submerged arc are the welding sources with high concentrated energy. A mathematical models of these welding sources may be introduced in class of piecewise continuous function with an efficient accuracy.