Relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices

Abstract

This paper is devoted to analysis of relative controllability of semilinear delay differential systems with linear parts defined by permutable matrices. By introducing a notion of delay Grammian matrix, we give a sufficient and necessary condition to examine that a linear delay controlled system is relatively controllable, which is a generalized criterion for the classical linear controlled systems without delay. Thereafter, we construct a suitable control function for semilinear delay controlled system, which admits us to following the framework of fixed point methods to consider the same issue. More precisely, we apply Krasnoselskii’s fixed point theorem to derive a relative controllability result for semilinear delay controlled systems. Finally, two numerical examples are presented to illustrate our theoretical results with the help of computing the desired control functions and inverse of delay Grammian matrix as well.

Introduction

Controllability issues for the classical linear/semilinear time-independent differential controlled systems in finite/infinite dimensional spaces have been addressed well. However, it will achieve a much more complicated situation for the same issues for time-independent delay differential controlled systems, even for linear differential system with pure delay with one input. In fact, in contrast with the theory of classical linear differential controlled systems, when driving these delay systems to rest, one is required not only to control the value of the state at the final time but also the memory accumulated with an after effect having that the initial functional condition introduces. In addition, the representation of solution is not easy to be characterized without knowing the fundamental matrix of a homogeneous delay differential system. As a result, various classes of control methods for delay differential systems are considered and different variants of controllability are developed in the past decades. Fortunately, in the past decade, there is a rapid development on representation of solution, which leads to results on asymptotic stability, finite-time stability and control problems for one/two order linear continuous/discrete delay systems. For more details, one can refer to [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] and reference therein.

We would like to focus on some meaningful contributions for controllability of problems for one order linear delay continuous/discrete systems. Recently, Khusainov and Shuklin [3] initially develop the classical ideas to seeking the fundamental matrix of $\dot{x}\left(t\right)=Bx(t-\tau ),t\ge 0,\tau \ge 0,$$B\in {\mathbb{R}}^{n\times n},$ and introduce a notation of so called delayed exponential matrix $e_{\tau }^{B·}$ (see Definition 2.1) to obtain a Kalman criterion to examine relative controllability in linear differential system with pure delay of the form

$$\begin{array}{c}\hfill \{\begin{array}{cc}& \dot{x}\left(t\right)=Bx(t-\tau )+bu\left(t\right),t\in J,\tau \ge 0,\hfill \\ & x\left(t\right)=\phi \left(t\right),-\tau \le t\le 0,\hfill \end{array}\end{array}$$

where $x:[-\tau ,t_1]\to {\mathbb{R}}^n$ is continuous differentiable on $J=[0,t_1],$ t₁ > 0, b is a constant vector, $u:J\to \mathbb{R}$ is an input and $\phi :[-\tau ,0]\to {\mathbb{R}}^n$ is a given function. System (1) is called relatively controllable, if for an arbitrary φ, the terminal state $x_1\in {\mathbb{R}}^n$ with finite t₁ > 0, there exists a control u*(t) such that

$$\begin{array}{c}\hfill \{\begin{array}{cc}& \dot{x}\left(t\right)=Bx(t-\tau )+b{u}^{*}\left(t\right),t\in J,\tau \ge 0,\hfill \\ & x\left(t\right)=\phi \left(t\right),-\tau \le t\le 0,\hfill \end{array}\end{array}$$

has a solution x(t, u*) ≔ x*(t) satisfying $x^{*}\left(t\right)=\phi \left(t\right)$ and $x^{*}\left(t_1\right)=x_1$.

For system (1) with the end time $t_1>(n-1)\tau$ ($n=\left[\frac{t}{\tau }\right]+1$), to be relative controllable iff $\det S_n\ne 0,S_n=\{b,Bb,\cdots ,B^{n-1}b\}$ (see [3, Theorem 4]) and the control function can be selected by virtue of delayed exponential matrix $e_{\tau }^{B·}$ (see Definition 2.1) of the formula

$$\begin{array}{c}\hfill u\left(t\right)=b^{\top }{e}_{\tau }^{B^{\top }(t_1-\tau -t)}{\left[{\int }_0^{t_1}{e}_{\tau }^{B(t_1-\tau -s)}b{b}^{\top }{e}_{\tau }^{B^{\top }(t_1-\tau -s)}ds\right]}^{-1}\eta ,\end{array}$$

where the symbol ⊤ denotes the transpose of the matrix and

$$\eta =x_1-e_{\tau }^{B{t}_1}\phi (-\tau )-{\int }_{-\tau }^0{e}_{\tau }^{B(t_1-\tau -s)}{\phi }^{\prime }\left(s\right)ds,$$

and the vector $x_1\in {\mathbb{R}}^n$ is pre-fixed.

Meanwhile, Diblík et al. [8], [9] transfer this idea to represent the solution of discrete delayed system by constructing discrete delayed exponential matrix and study controllability of linear discrete pure delay systems with constant coefficients and one control function [10], [11]. Moreover, Pospíšil et al. [22] extend the study of controllability to linear discrete pure delay systems with constant coefficients and multiple control functions and derive representation of solution in a form of a matrix polynomial using the Z-transform [23] to a system of nonhomogeneous linear difference equations with any finite number of constant delays and linear parts given by pairwise permutable matrices. Various criterions of relative controllability for linear discrete pure delay systems are presented and the associated control functions are also constructed.

A much more complicated situation occurs if we deal with relative controllability of the following semilinear delay differential systems with linear parts defined by pairwise permutable matrices:

$$\begin{array}{c}\hfill \{\begin{array}{cc}& \dot{x}\left(t\right)=Ax\left(t\right)+Bx(t-\tau )+f(t,x\left(t\right))+Cu\left(t\right),t\in J,\tau \ge 0,\hfill \\ & x\left(t\right)=\phi \left(t\right),-\tau \le t\le 0,\hfill \end{array}\end{array}$$

where $A,B,C\in {\mathbb{R}}^{n\times n},$$AB=BA,$$f:J\times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $\phi \in C^1([-\tau ,0],{\mathbb{R}}^n)$. The state x( · ) takes values from ${\mathbb{R}}^n$ and the control function u( · ) takes value from $L^2(J,{\mathbb{R}}^n)$.

Concerning relative controllability of system (3), we would like to address the difficulties as follows:

• The formula of solution of system (3) is much more complicated since the representation of solution involving a more complicated delayed matrix exponential $e_{\tau }^{e^{-A\tau }Bt},t\in J$.

• Since the both linear parts A and B are appeared, it is debatable whether we can establish the similar Kalman criterion since the Cayley formula for delayed matrix exponential $e_{\tau }^{e^{-A\tau }Bt},t\in J$ is not an easy task.

Except using Kalman criterion to to judge whether linear controlled system is controllable, Grammian matrix criterion is also an alternative method, which has been widely used to deal with linear controlled systems. The controllability problem for the following linear differential systems with control

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \{\begin{array}{cc}& \dot{x}\left(t\right)=Ax\left(t\right)+Cu\left(t\right),t\in J,\hfill \\ & x\left(0\right)=x_0\in {\mathbb{R}}^n,\hfill \end{array}\end{array}\end{array}$$

is known to be solved iff the Grammian matrix of the form

$$\begin{array}{c}\hfill W_c[0,t_1]={\int }_0^{t_1}{e}^{-As}C{C}^{\top }{e}^{A^{\top }s}ds.\end{array}$$

is non-singular.

Now we revisit relative controllability of system (1). By checking the proof of [3, Theorem 4] via the control function (2), one can directly conclude that a sufficient condition for system (1) is relatively controllable when the following Grammian type matrix

$${\int }_0^{t_1}{e}_{\tau }^{B(t_1-\tau -s)}b{b}^{\top }{e}_{\tau }^{B^{\top }(t_1-\tau -s)}ds$$

is non-singular. This implies that it is possible to adopt Grammian type matrix method to address relative controllability of system (3) although it is not easy to verify the necessity.

The main contributions are stated as follows:

• Concerning on linear delay differential controlled system, instead of seeking Kalman type criterion, we introduce a notion of delay Grammian matrix and display the relationship between non-singular property of delay Grammian matrix between relative controllability linear delay controlled system. Meanwhile, we also give an algorithm for constructing a control.

• Except for a criterion of relative controllability for linear delay differential controlled system we construct a suitable control function for semilinear delay differential controlled system and adopt the framework of fixed point methods to derive relative controllability via Krasnoselskii’s fixed point theorem as well.

The rest of this paper is organized as follows. In Section 2, we give some necessary notations, concepts and lemmas. In Section 3, we firstly investigate relative controllability of linear delay controlled differential system and give a delay Grammian matrix criterion. Secondly, we turn relative controllability of semilinear delay controlled differential system into a fixed point problem, which admit us to utilize Krasnoselskii’s fixed point to check our main theorem. Two numerical examples are given in final section to demonstrate the application of our main results.