# Controllability of Semilinear Systems with Multiple Variable Delays in Control

## Abstract

**:**

## 1. Introduction

## 2. System Description

- $x\left(t\right)\in {R}^{n}$ is a pseudo or instantaneous state vector,
- $u\in {L}_{loc}^{2}\left(\left[{t}_{0},\infty \right),{R}^{p}\right)$ is an admissible control,
- A is (n × n)-dimensional constant matrix with real elements,
- B
_{i}are (n × m)-dimensional constant matrices with real elements for i = 0, 1, 2, …, M, - f is the continuous mapping $f:{R}^{n}\times {R}^{p}\times {R}^{p}\times \dots \times {R}^{p}\times \dots \times {R}^{p}\to {R}^{n}$

_{i}(t):[t

_{0},t

_{1}]→R, i = 0, 1, 2, …, M, represent deviating arguments in the admissible controls and in the state variables, i.e., v

_{i}(t) = t − h

_{i}(t), where h

_{i}(t) are lumped time varying delays for i = 0, 1, 2, …, M. Moreover, v

_{i}(t) ≤ t for t ∈ [t

_{0},t

_{1}], and i = 0, 1, 2, 3, …, M.

_{i}(t):[v

_{i}(t

_{0}),v

_{i}(t

_{1})]→[t

_{0},t

_{1}], i = 0, 1, 2, 3, …, M, such that r

_{i}(v

_{i}(t)) = t for t ∈ [t

_{0},t

_{1}]. Furthermore, only for simplicity and compactness of notations, let us assume that v

_{0}(t) = t and for a given time t

_{1}the functions v

_{i}(t) satisfy the following inequalities.

_{M}(t

_{1}) ≤ v

_{M−1}(t

_{1}) ≤ … ≤ v

_{m+1}(t

_{1}) ≤ t

_{0}= v

_{m}(t

_{1}) < v

_{m−1}(t

_{1}) ≤ … ≤ v

_{1}(t

_{1}) ≤ v

_{0}(t

_{1}) = t

_{1}

_{t}(s)}, where u

_{t}(s) = u(s) for s$\in \left[{v}_{M}\left(t\right),t\right)$. Moreover, it should be pointed out, that only the complete state z(t) completely describes the behavior of the control system at a given time t.

_{0}− v

_{M}(t

_{0}) > 0 be given. For a given function x:[t

_{0}− h,t

_{1}]→R

^{n}and t ∈ [t

_{0},t

_{1}], the symbol x

_{t}usually denotes the function on [−h,0] defined by x

_{t}(s) = x(t + s) for s ∈ [−h,0].

_{M}(t

_{0}),t

_{1}]→R

^{p}, and t ∈ [t

_{0},t

_{1}], the symbol u

_{t}denotes the function on [v

_{M}(t),t) defined by the equality u

_{t}(s) = u(t + s) for s ∈ [v

_{M}(t),t). For example, ${u}_{{t}_{0}}$ is the initial control function defined on time interval [v

_{M}(t

_{0}),t

_{0}).

_{0},t

_{1}], the aim is to find an admissible control so that the instantaneous state x(t

_{1}) can be reached using this admissible control.

_{M}(t

_{1}),t

_{1}) should be a given m-dimensional function.

## 3. Preliminaries

**Rothe’s fixed-point theorem**[12,14]. Let E be a Banach space and B be a closed convex subset of E such that zero of E is contained in the interior of B. Let $g:B\to E$ be a continuous mapping with g(B), relatively compact in E and g(∂B) is a subset of ∂B, where ∂B denotes the boundary of B. Then, there is a point ${x}^{\ast}\in B$ such that $g\left({x}^{\ast}\right)={x}^{\ast}$.

_{0},t

_{1}], the aim is to find an admissible controls such that the instantaneous state x(t

_{1}) can be reached using this admissible control.

_{M}(t

_{1}),t

_{1}] should be a given m-dimensional function.

**Definition**

**1**

**.**The attainable set at time t

_{1}> t

_{0}from the given initial complete state $z\left({t}_{0}\right)=\{x\left({t}_{0}\right),{u}_{{t}_{0}})$ for the time delay control system (1) is the set as

_{0}) = {x(t

_{0}),${u}_{{t}_{0}}$).

**Definition**

**2.**

_{0,}t

_{1}] if for each initial complete state z(t

_{0}) = {x(t

_{0}),${u}_{{t}_{0}}$) and in each final instantaneous state x

_{1}∈ R

^{n}, there exists an admissible control${u}_{1}\in {L}^{2}\left(\left[{t}_{0},{t}_{1}\right],{R}^{p}\right)$such that x(t

_{1},u

_{1}) = x

_{1}.

_{0,}t

_{1}] if the attainable set K([t

_{0,}t

_{1}]) is the whole space R

^{n}.

_{i}

_{+1}(t

_{1}), v

_{i}

_{+1}(t

_{1})), i = 0, 1, 2, …, m − 1.

^{n}

_{1}, ${u}_{{t}_{0}}$) ∈ R

^{n}.

_{m}(t,s) is an n × p dimensional matrix.

## 4. Controllability Conditions

_{0}(t

_{0},t

_{1}) for the linear dynamical control system (5):

_{0},t

_{1}) is the n × n dimensional symmetric matrix depending only on time interval [t

_{0},t

_{1}] and system parameters.

**Theorem**

**1**

_{0},t

_{1}) = n

_{u}(t) is the solution of Equation (1) for the control u(t).

_{f}(u) = G(u) + H(u)

_{0}(t

_{0},t

_{1}) and, using Rothe’s fixed-point theorem, the following sufficient condition for relative controllability on [t

_{0},t

_{1}] may be formulated and proved.

**Theorem**

**2.**

_{0},t

_{1}] and the following inequality holds

_{0,}t

_{1}].

_{0}) to a given final instantaneous state x

_{1}= x(t

_{1}) at time t

_{1}> t

_{0}is given by the formula

**Proof.**

_{0}) denote the ball centered at zero and with radius d

_{0}> 0 and boundary ∂B(0,d

_{0}). Then, from the above inequality, it follows that P(∂B(0,d

_{0})) $\in $ ∂B(0,d

_{0}). Thus, nonlinear operator P is a compact operator and maps the sphere ∂B(0,d

_{0}) into the interior of the ball ∂B(0,d

_{0}).

_{0}) = {x(t

_{0}),${u}_{{t}_{0}}$) to a final instantaneous state x

_{1}= x(t

_{1}) at time t

_{1}> t

_{0}is given by the following formula

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Klamka, J. Controllability of Dynamical Systems; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1991. [Google Scholar]
- Klamka, J. Controllability of dynamical systems. A survey. Bull. Pol. Acad. Sci. Tech. Sci.
**2013**, 61, 335–342. [Google Scholar] [CrossRef] - Mahmudov, N.I.; Denker, A. On controllability of linear stochastic systems. Int. J. Control
**2000**, 73, 144–151. [Google Scholar] [CrossRef] - Mahmudov, N.I.; Zorlu, S. Controllability of nonlinear stochastic systems. Int. J. Control
**2003**, 76, 95–104. [Google Scholar] [CrossRef] - Zhou, H.X. Controllability properties of linear and semilinear abstract control systems. SIAM J. Control Optim.
**1984**, 22, 405–422. [Google Scholar] [CrossRef] - Capot, D.; Ghita, M.; Ionescu, C. Simple alternative to PID-type control for processes with variable Time Delay. Processes
**2019**, 7, 146. [Google Scholar] [CrossRef][Green Version] - Balachandran, K.; Dauer, J.P. Controllability of nonlinear systems via fixed point theorems. J. Optim. Theory Appl.
**1987**, 53, 345–352. [Google Scholar] [CrossRef] - Balachandran, K.; Dauer, J.P. Controllability of nonlinear systems in Banach spaces: A Survey. J. Optim. Theory Appl.
**2002**, 115, 7–28. [Google Scholar] [CrossRef] - Sakthivel, R.; Mahmudov, N.I.; Kim, H.J. On controllability of second-order nonlinear impulsive differential systems. Nonlinear Analysis. Theory Methods Appl.
**2009**, 71, 45–52. [Google Scholar] [CrossRef] - Anguraj, A.; Ravikumar, K.; Baleanu, D. Approximate controllability of semilinear impulsive stochastic system with nonlocal condition and Poisson jumps. Adv. Differ. Equ.
**2020**, 2020, 1–13. [Google Scholar] [CrossRef][Green Version] - Naito, K. Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim.
**1987**, 25, 715–722. [Google Scholar] [CrossRef] - Leiva, H. Rothe’s fixed point theorem and controllability of semilinear nonautonomous systems. Syst. Control Lett.
**2014**, 67, 14–18. [Google Scholar] [CrossRef] - Park, J.Y.; Balachandran, K.; Arthi, G. Controllability of impulsive neutral integrodifferential systems with infinite delay in Banach spaces. Nonlinear Anal. Hybrid Syst.
**2009**, 3, 184–194. [Google Scholar] [CrossRef] - Sikora, B. On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays. Kybernetika
**2019**, 55, 675–689. [Google Scholar] [CrossRef][Green Version] - Klamka, J. Controllability and minimum energy control. In Studies in Systems, Decision and Control; Springer: New York, NY, USA, 2018; Volume 162. [Google Scholar]
- Sikora, B.; Klamka, J. Constrained controllability of fractional linear systems with delays in control. Syst. Control. Lett.
**2017**, 106, 9–15. [Google Scholar] [CrossRef] - Balachandran, K.; Park, D.; Manimegalai, P. Controllability of second-order integrodifferential evolution systems in Banach spaces. Comput. Math. Appl.
**2005**, 49, 1623–1642. [Google Scholar] [CrossRef][Green Version] - Peichl, G.; Schappacher, W. Constrained controllability in Banach spaces. SIAM J. Control Optim.
**1986**, 24, 1261–1275. [Google Scholar] [CrossRef] - Sikora, B.; Klamka, J. Cone-type constrained relative controllability of semilinear fractional linear systems with delays. Kybernetika
**2017**, 53, 370–381. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Klamka, J. Controllability of Semilinear Systems with Multiple Variable Delays in Control. *Mathematics* **2020**, *8*, 1955.
https://doi.org/10.3390/math8111955

**AMA Style**

Klamka J. Controllability of Semilinear Systems with Multiple Variable Delays in Control. *Mathematics*. 2020; 8(11):1955.
https://doi.org/10.3390/math8111955

**Chicago/Turabian Style**

Klamka, Jerzy. 2020. "Controllability of Semilinear Systems with Multiple Variable Delays in Control" *Mathematics* 8, no. 11: 1955.
https://doi.org/10.3390/math8111955