On "Equation missing" -equivalence of impulsive differential equations and its applications to partial impulsive differential equations

Abstract

By means of the Schauder-Tychonoff principle, L p (k)-equivalence are established between linear and nonlinear perturbed impulsive differential equations with an unbounded linear part in an arbitrary Banach space. The feasibility of our theoretical results is illustrated by an example involving partial impulsive differential equations of the parabolic type.

On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls

“…Problem 3. Find absolutely continuous function c( ) and discrete controld( ) almost everywhere satisfying system (10) and conditions…”

“…Returning in the first component of functionc (4n) to the initial variable c by formulas (21), (18), (15), we get the known function c( ). Due to design of the systems (63), (27), and (22), function c( ) satisfies to system (10), given that in the right side of system (10) a known functiond( ) is substituted. According to (74), (51), (24), (21), (20), (18), (17), and (15), the function c( ) satisfies to boundary condition (12).…”

“…5. The solution of the Cauchy problem for system (10), (25) with initial data (12), (26) after the substitution of function d( ) into its right-part side of first n equations is found in Section 4. As a result, we obtain absolutely continuous function c( ) and piecewise constant controld( ), which is the solution of Problem 3.…”

“…Due to this circumstance, studying more complicated types of BVPs in the class of piecewise constant control functions and relay (bang‐bang) controls becomes quite important. The study of BVPs at a finite interval of time in the mentioned class of controls includes questions related to finding necessary and sufficient conditions that guarantee the transfer of a controlled system from the initial state to a given final state with the study and assessment of the reachable domain as well as the development of exact or approximate methods for constructing control functions for which ODE systems solution connect certain points in the phase space . To the difficulties of research questions mentioned above, there are added questions related to finding the switching points of the desired piecewise constant control.…”