Robustness of the controllability for the heat equation under the influence of multiple impulses and delays

LaSalle wrote the following: “it is never possible to start the system exactly in its equilibrium state, and the system is always subject to outside forces not taken into account by the differential equations. The system is disturbed and is displaced slightly from its equilibrium state. What happens? Does it remain near the equilibrium state? This is stability. Does it remain near the equilibrium state and in addition tend to return to the equilibrium? This is asymptotic stability.” Continuing with what LaSalle said, we conjecture that real-life systems are always under the influence of impulses, delays, memory, nonlocal conditions, and noises, which are intrinsic phenomena no taken into account by the mathematical model that is representing by a differential equation. For many control systems in real life, delays, impulses, and noises are natural properties that do not change their behavior. So, we conjecture that, under certain conditions, the abrupt changes, delays, and noises as perturbations of a system do not modify certain properties such as controllability. In this regard, we prove the interior S ∗ -controllability of the semilinear stochastic heat equation with impulses and delay on the state variable, and this is done by using new techniques avoiding fixed point theorems employed by Bashirov et al.

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“…erefore, system (1) can be written as abstract functional differential equations with impulses and noses (see Acosta-Leiva [33].…”

Section: Abstract Formulation Of the Problemmentioning

confidence: 99%

Controllability of Impulsive Semilinear Stochastic Heat Equation with Delay

Leiva

Narvaez

Sivoli

2020

International Journal of Differential Equations

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“…In particular, it avoids fixed‐point theorems. Previously, this method was successfully applied in the context of approximate controllability 22–24, 36–37 . We modify it to the context of exact controllability, which allows to prove that a semilinear system in a Hilbert space, which satisfies general conditions for the existence of its solution, steers any initial state to any final state from the domain

D false(A false)

of the system operator

A

if its linear part is exactly controllable at every noninitial time moment, the nonlinear term is bounded, and the norm of the inverse of the Gramian does not increase as time goes to zero faster than the reciprocal function, that is,

t false‖ Q {false(t false)}^{- 1} false‖

is bounded for

0 < t \leq T

, where

Q false(t false)

is the respective Gramian.…”

Section: Introductionmentioning

confidence: 99%

“…[9][10][11][12][13] The issue is different for nonlinear systems. Here, the investigations are directed to proving sufficient conditions of controllability for first order, [14][15][16][17] second order, [18][19][20] fractional order, 21 impulsive, [22][23][24] constrained, 25 with memory, 26 and other kinds of systems. Different methods are applied for investigation of controllability such as fixed-point theorems, [27][28][29][30][31][32] Hilbert uniqueness method by Lions, 33 and variational method.…”

Section: Introductionmentioning

confidence: 99%

On exact controllability of semilinear systems

Bashirov

2020

Math Methods in App Sciences

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In this paper, we suggest a proof method for the exact controllability of semilinear systems. Previously, it was applied in the context of the approximate controllability. In comparison to the traditional proof methods, the suggested method is easy because it does not require heavy estimations. Also, it is effective because it does not need some unnatural conditions, which are required for application of fixed‐point theorems. Using these advantages of the suggested method, we prove that a semilinear system in a Hilbert space, which satisfies general conditions for the existence of its solution, steers any initial state to any final state from the domain of the system operator if its linear part is exactly controllable at every noninitial time moment, the nonlinear term is bounded, and the norm of the inverse of the Gramian increases as time goes to zero not faster than the reciprocal function. The obtained result is demonstrated on examples.

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