## 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

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