On Partial Complete Controllability of Semilinear Systems

Abstract

Many control systems can be written as a first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples.

1. Introduction

A concept of controllability, defined by Kalman [1] in 1960 for finite dimensional control systems, is a property of attaining every point in the state space from every initial state point for a finite time. Further studies on this concept in infinite dimensional spaces demonstrated that it is suitable to consider its two versions: a stronger version of complete controllability and a weaker version of approximate controllability. The reason for these versions was the fact that many infinite dimensional control systems are not completely controllable while they are approximately controllable (see Fattorini [2] and Russell [3]). The necessary and sufficient conditions for complete and approximate controllability concepts are almost completely studied and presented in, for example, Curtain and Zwart [4], Bensoussan [5], Bensoussan et al. [6], Zabczyk [7], Bashirov [8], Klamka [9], and so forth for linear systems; Balachandran and Dauer [10, 11], Klamka [12], Mahmudov [13], Li and Yong [14], and so forth for nonlinear systems; Sakthivel et al. [15–17], Yan [18], and so forth for fractional differential systems; and Ren et al. [19] for differential inclusions.

Recently, in Bashirov et al. [20, 21] the partial controllability concepts were initiated. The idea of these concepts is that some control systems, including higher order differential equations, wave equations, and delay equations, can be written as a first-order differential equation only by enlarging the dimension of the state space. Therefore, the theorems on controllability, which are formulated for control systems in the form of first-order differential equation, are too strong for them because they involve the enlarged state space. In such cases the partial controllability concepts became preferable, which assume the original state space. The basic controllability conditions for linear systems, including resolvent conditions from Bashirov and Mahmudov [22] and Bashirov and Kerimov [23] (see also [24–26]), are extended to partial controllability concepts by just a replacement of the controllability operator by its partial version.

In this paper our aim is to study the partial complete controllability of semilinear systems. The controllability concepts for semilinear systems are intensively discussed in the literature (see Balachandran and Dauer [10, 11], Klamka [12], Mahmudov [13], Sakthivel et al. [15, 17], and references therein). A basic tool of study in these works is fixed point theorems. In this paper, we also use one of the fixed point theorems, a contraction mapping theorem, and find a sufficient condition for the partial complete controllability of a semilinear control system.

The rest of this paper is organised in the following way. In Section 2 we set the problem, give basic definitions, and motivate the partial controllability concepts by considering a higher order differential equation, a wave equation, and a delay equation. Section 3 contains the proof of the main result. In Section 4, we demonstrate the main result in the examples. Finally, Section 5 contains directions of further research regarding partial controllability concepts.

2. Setting the Problem and Motivation

Consider the basic semilinear control system 𝑥^′_𝑡=𝐴𝑥_𝑡+𝐵𝑢_𝑡+𝑓𝑡,𝑥_𝑡,𝑢_𝑡,0<𝑡≤𝑇,𝑥₀∈𝑋, (1) on the interval [0,𝑇] with 𝑇>0, where 𝑥 and 𝑢 are state and control processes. We assume that the following conditions hold. (A)𝑋and 𝑈 are separable Hilbert spaces, 𝐻 is a closed subspace of 𝑋, and 𝐿 is a projection operator from 𝑋 to 𝐻; (B)𝐴 is a densely defined closed linear operator on 𝑋, generating a strongly continuous semigroup 𝑒^𝐴𝑡, 𝑡≥0; (C)𝐵 is a bounded linear operator from 𝑈 to 𝑋; (D)𝑓 is a nonlinear function from [0,𝑇]×𝑋×𝑈 to 𝑋, satisfying that(i)𝑓 is continuous on [0,𝑇]×𝑋×𝑈; (ii)𝑓 is Lipschitz continuous with respect to 𝑥 and 𝑢 that is, for all 𝑡∈[0,𝑇], 𝑢,𝑣∈𝑈 and 𝑥,𝑦∈𝑋, 𝑓(𝑡,𝑥,𝑢)−𝑓𝑡,𝑦,𝑣≤𝐾𝑥−𝑦+‖𝑢−𝑣‖ (2) for some 𝐾≥0; (E)𝑈_ad=𝐶(0,𝑇;𝑈) is the space of all continuous functions from [0,𝑇] to 𝑈. Define the controllability and 𝐿-partial controllability operators 𝑄_𝑡 and 𝑄_𝑡 by 𝑄_𝑡=^𝑡₀𝑒^𝐴𝑠𝐵𝐵^∗𝑒^𝐴∗𝑠𝑑𝑠,𝑄_𝑡=𝐿𝑄_𝑡𝐿^∗, (3) where 𝐿^∗ is the adjoint of 𝐿. The 𝐿-partial controllability operator becomes the controllability operator if 𝐿=𝐼 (the identity operator). We will also assume the following condition;(F)𝑄_𝑇 is coercive; that is, there is 𝛾>0 such that ⟨𝑄_𝑇ℎ,ℎ⟩≥𝛾‖ℎ‖² for all ℎ∈𝐻.

Note that this condition implies the existence of 𝑄_𝑇⁻¹ as a bounded linear operator and ‖𝑄_𝑇⁻¹‖≤1/𝛾. Respectively, the linear system associated with (1) (the case when 𝑓=0) is 𝐿-partially complete controllable on the interval [0,𝑇] (see, Bashirov et al. [20, 21]).

The above conditions imply the existence of a unique continuous solution of (1) in the mild sense for every 𝑢∈𝑈_ad and 𝑥₀∈𝑋 (see Li and Yong [14] and Byszewski [27]); that is, there is a unique continuous function 𝑥 from [0,𝑇] to 𝑋 such that 𝑥_𝑡=𝑒^𝐴𝑡𝑥₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝐵𝑢_𝑠+𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠. (4)

Let 𝐷^𝑥0_𝑇=ℎ∈𝐻∶∃𝑢∈𝑈_adsuchthatℎ=𝐿𝑥_𝑇. (5) Following Bashirov et al. [21], the semilinear control system (1) is said to be 𝐿-partially complete controllable on 𝑈_ad if 𝐷^𝑥0_𝑇=𝐻 for all 𝑥₀∈𝑋. Similarly, the semilinear system in (1) is said to be 𝐿-partially approximate controllable on 𝑈_ad if 𝐷^𝑥0_𝑇=𝐻 for all 𝑥₀∈𝑋, where 𝐷^𝑥0_𝑇 is the closure of 𝐷^𝑥0_𝑇. If 𝐻=𝑋, these are just well-known complete and approximate controllability concepts, respectively. In this paper, we study the concept of 𝐿-partial complete controllability.

The reason for studying 𝐿-partial controllability concepts is that many systems can be written in the form of (1) if the original state space is enlarged. Therefore, suitable controllability concepts for such systems are the 𝐿-partial controllability concepts with the operator 𝐿 projecting the enlarged state space to the original one. Here are some examples of such systems, which are discussed in Bashirov [8], Section  3.1.1, in more details.

Example 1. Consider the system 𝑥_𝑡^(𝑛)=𝑓𝑡,𝑥_𝑡,𝑥^′_𝑡,…,𝑥_𝑡^(𝑛−1),𝑢_𝑡, (6) assuming that its state space is the one-dimensional space ℝ. The ordinary controllability concepts for this system are the equality to or denseness in ℝ of the respective attainable set. We can write this system as the first-order differential equation 𝑦^′_𝑡=𝐴𝑦_𝑡+𝐹𝑡,𝑦_𝑡,𝑢_𝑡 (7) if 𝑦_𝑡=𝑥_𝑡𝑥^′_𝑡⋮𝑥_𝑡^(𝑛−2)𝑥_𝑡^(𝑛−1),𝐹=00⋮0𝑓,𝐴=01⋯0000⋯00⋮⋮⋱⋮⋮00⋯0100⋯00𝑡,𝑦,𝑢𝑡,𝑥,𝑥^′,…,𝑥^(𝑛−1).,𝑢 (8) The state space of this system is the 𝑛-dimensional Euclidean space ℝ^𝑛 and, respectively, its attainable set is a subset of ℝ^𝑛. Therefore, the controllability concepts of the system for 𝑦 are stronger than those of the system for 𝑥. But if we define the projection operator 𝐿 by 𝐿=10⋯00∶ℝ^𝑛⟶ℝ, (9) then the 𝐿-partial controllability concepts of the system for 𝑦 become the same as the ordinary controllability concepts of the system for 𝑥.

Example 2. Consider the nonlinear wave equation 𝜕²𝑥_𝑡,𝜃𝜕𝑡²=𝜕²𝑥_𝑡,𝜃𝜕𝜃²+𝑓𝑡,𝑥_𝑡,𝜃,𝜕𝑥_𝑡,𝜃𝜕𝑡,𝑢_𝑡, (10) where 𝑥 is a real-valued function of two variables 𝑡≥0 and 0≤𝜃≤1. The state space of this system is 𝐿₂(0,1) (the space of square integrable functions on [0,1]). This system can be written as the first-order abstract differential equation 𝑦^′_𝑡=𝐴𝑦_𝑡+𝐹𝑡,𝑦_𝑡,𝑢_𝑡 (11) if 𝑦_𝑡=𝑥_𝑡,𝜃𝜕𝑥_𝑡,𝜃𝑑𝜕𝑡,𝐴=0𝐼₂𝑑𝜃²0,𝐹=0𝑓𝑡,𝑦,𝑢𝑡,𝑦₁,𝑦₂,,𝑢 (12) where 𝑦∈𝐿₂(0,1)×𝐿₂(0,1). The state space 𝐿₂(0,1)×𝐿₂(0,1) of the system for 𝑦 is the enlargement of the state space 𝐿₂(0,1) of the system for 𝑥. This is a cost that is paid to bring the wave equation to the form of first-order differential equation. The ordinary controllability concepts for the system (11) are too strong for the system (10). If 𝐿=𝐼0∶𝐿₂(0,1)×𝐿₂(0,1)⟶𝐿₂(0,1), (13) then 𝐿-partially controllability concepts of the system for 𝑦 become ordinary controllability concepts of the system for 𝑥.

Example 3. Consider the system 𝑥^′_𝑡=𝑓𝑡,𝑥_𝑡,⁰_−𝜀𝑥_𝑡+𝜃𝑑𝜃,𝑢_𝑡, (14) which contains a simple distributed delay in the nonlinear term, assuming that 𝑥 is a real-valued function. Then the state space is ℝ. To bring this system to a system without delay, enlarge ℝ to ℝ×𝐿₂(−𝜀,0) and define 𝐿₂(−𝜀,0)-valued function 𝑥_𝑡𝜃=𝑥_𝑡+𝜃,𝑡≥0,−𝜀≤𝜃≤0. (15) Then for 𝑦_𝑡=𝑥_𝑡𝑥_𝑡0𝑑,𝐴=00,𝐹=𝑓(𝑑𝜃𝑡,𝑦,𝑢𝑡,𝑥,)0,𝑥,𝑢 (16) the above system can be written as the abstract system 𝑦^′_𝑡=𝐴𝑦_𝑡+𝑓𝑡,𝑦_𝑡,𝑢_𝑡. (17) Similar to the previous examples, one can easily observe that the ordinary controllability concepts for the system (17) are too strong for the system (14), but the 𝐿-partial controllability concepts of the system for 𝑦 with 𝐿=𝐼0∶ℝ×𝐿₂(0,1)⟶ℝ (18) are exactly the ordinary controllability concepts of the system for 𝑥.

These examples motivate a study of the partial controllability concepts. In this paper it is proved that under the conditions (A)–(F), the system in (1) is 𝐿-partially complete controllable.

3. Main Result

Denote 𝑋=𝐶(0,𝑇;𝑋). Then 𝑋×𝑈_ad is a Banach space with the norm ‖(⋅,⋅)‖𝑋_×𝑈ad=‖⋅‖𝑋+‖⋅‖_𝑈ad. (19)

Lemma 4. Under the conditions (A), (B), and (C), 𝑄_𝑡≤𝑄_𝑇,𝑄_𝑡≤𝑄_𝑇,0≤𝑡≤𝑇. (20)

Proof. It is easy to see that 𝑄_𝑡=𝑄^∗_𝑡 and ⟨𝑄_𝑡𝑥,𝑥⟩≥0 for all 𝑥∈𝑋. Hence, 𝑄_𝑡=sup_‖𝑥‖=1⟨𝑄_𝑡𝑥,𝑥⟩. (21) Then 𝑄_𝑇𝑥,𝑥=^𝑇₀⟨𝑒^𝐴𝑠𝐵𝐵^∗𝑒^𝐴∗𝑠𝑥,𝑥⟩𝑑𝑠=⟨𝑄_𝑡𝑥,𝑥⟩+^𝑇_𝑡⟨𝑒^𝐴𝑠𝐵𝐵^∗𝑒^𝐴∗𝑠𝑥,𝑥⟩𝑑𝑠=⟨𝑄_𝑡𝑥,𝑥⟩+^𝑇_𝑡𝐵^∗𝑒^𝐴∗𝑠𝑥²≥𝑄𝑑𝑠_𝑡𝑥,𝑥. (22) This implies ‖𝑄_𝑡‖≤‖𝑄_𝑇‖. The conclusion of the lemma regarding 𝑄_𝑇 follows from ⟨𝑄_𝑡𝑥,𝑥⟩=⟨𝑄_𝑡𝐿^∗𝑥,𝐿^∗𝑥⟩.

The proof of the following lemma appears in different forms in several papers, for example, Mahmudov [13]. Our proof is a minor modification of them.

Lemma 5. Assume that the conditions (A)–(F) hold and take arbitrary ℎ∈𝐻. Then for the operator 𝐺∶𝑋×𝑈_𝑎𝑑→𝑋×𝑈_𝑎𝑑, defined by 𝐺𝑦,𝑣(𝑡)=(𝑌(𝑡),𝑉(𝑡)),0≤𝑡≤𝑇, (23) where 𝑌(𝑡)=−𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑦_𝑠,𝑣_𝑠+𝑑𝑠_𝑡0𝑒^{𝐴(𝑡−𝑠)}𝑓𝑠,𝑦_𝑠,𝑣_𝑠𝑉(𝑡)𝑑𝑠,=𝐵^∗𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹×ℎ−𝐿𝑒^𝐴𝑇𝑥₀−𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑦_𝑠,𝑣_𝑠𝑑𝑠, (24) the following inequality holds: 𝐺(≤𝑄𝑦,𝑣−𝐺𝑧,𝑤)1+^𝑇𝑀𝛾+‖𝐵‖𝑀𝛾×𝑀𝐾𝑇𝑦−𝑧+‖𝑣−𝑤‖, (25) where 𝑀=sup_{0≤𝑡≤𝑇}𝑒^𝐴𝑡. (26)

Proof. Let (𝑦,𝑣) and (𝑧,𝑤) be two functions in 𝑋×𝑈_ad such that 𝐺(𝑦,𝑣)=(𝑌,𝑉) and 𝐺(𝑧,𝑤)=(𝑍,𝑊). Then, 𝑋𝐺(𝑦,𝑣)−𝐺(𝑧,𝑤)_×𝑈ad=‖‖𝑋𝑌−𝑍+‖𝑉−𝑊‖_𝑈ad. (27) Here, 𝑋‖𝑌−𝑍‖ can be estimated as follows: ‖𝑌−𝑍‖=max_{𝑡∈[0,𝑇]}^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝑓𝑠,𝑦_𝑠,𝑣_𝑠−𝑓𝑠,𝑧_𝑠,𝑤_𝑠𝑑𝑠−𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿×^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑦_𝑠,𝑣_𝑠−𝑓𝑠,𝑧_𝑠,𝑤_𝑠𝑑𝑠≤max_{[]𝑡∈0,𝑇}𝑀+𝑀²𝑄_𝑡𝑄_𝑇⁻¹×^𝑇₀𝑓𝑠,𝑦_𝑠,𝑣_𝑠−𝑓𝑠,𝑧_𝑠,𝑤_𝑠𝑄𝑑𝑠≤𝑀1+𝑀_𝑇𝑄_𝑇⁻¹×^𝑇₀𝑓𝑠,𝑦_𝑠,𝑣_𝑠−𝑓𝑠,𝑧_𝑠,𝑤_𝑠≤𝑄𝑑𝑠1+^𝑇𝑀𝛾𝑀𝐾^𝑇₀𝑦_𝑠−𝑧_𝑠+𝑣_𝑠−𝑤_𝑠≤𝑄𝑑𝑠1+^𝑇𝑀𝛾𝑀𝐾𝑇𝑦−𝑧+‖𝑣−𝑤‖. (28) Similarly, for ‖𝑉−𝑊‖_𝑈ad, we have ‖𝑉−𝑊‖=max_{𝑡∈[0,𝑇]}𝐵^∗𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿×^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑦_𝑠,𝑣_𝑠−𝑓𝑠,𝑧_𝑠,𝑤_𝑠𝑑𝑠≤𝑀²‖𝐵‖𝑄_𝑇^−1𝑇₀𝑓𝑠,𝑦_𝑠,𝑣_𝑠−𝑓𝑠,𝑧_𝑠,𝑤_𝑠𝑑𝑠≤𝑀²‖𝐵‖1𝛾𝐾^𝑇₀𝑦_𝑠−𝑧_𝑠+𝑣_𝑠−𝑤_𝑠≤‖𝐵‖𝑀𝑑𝑠𝛾𝑀𝐾𝑇𝑦−𝑧+‖𝑣−𝑤‖. (29) Combining (28) and (29), we obtain the demanded inequality.

Lemma 6. Under the conditions (A)–(F), if 𝑄1+^𝑇𝑀𝛾+‖𝐵‖𝑀𝛾𝑀𝐾𝑇<1, (30) then the operator 𝐺, mapping 𝑋×𝑈_𝑎𝑑 into 𝑋×𝑈_𝑎𝑑, has a unique fixed point (𝑥,𝑢)∈𝑋×𝑈_𝑎𝑑.

Proof. By Lemma 5, 𝐺 is a contraction mapping. Also, the space 𝑋×𝑈_ad is complete. Hence, 𝐺 has a fixed point.

Theorem 7. Under the conditions (A)–(F) and (30), the semilinear system (1) is 𝐿-partially complete controllable on [0,𝑇].

Proof. Take any 𝑥₀∈𝑋 and ℎ∈𝐻. Show that there is 𝑢∈𝑈_ad such that ℎ=𝐿𝑥_𝑇. To this end, consider 𝑢, defined as follows: 𝑢_𝑡=𝐵^∗𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹×ℎ−𝐿𝑒^𝐴𝑇𝑥₀−𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠. (31) Substituting (31) in (4) and applying Fubini’s theorem (see Bashirov [8], p. 45), we obtain 𝑥_𝑡=𝑒^𝐴𝑡𝑥₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝐵𝐵^∗𝑒^{𝐴∗(𝑡−𝑠)}𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹ℎ−𝐿𝑒^𝐴𝑇𝑥₀𝑑𝑠−^𝑡₀^𝑇₀𝑒^{𝐴(𝑡−𝑠)}𝐵𝐵^∗𝑒^{𝐴∗(𝑡−𝑠)}𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿𝑒^{𝐴(𝑇−𝑟)}×𝑓𝑟,𝑥_𝑟,𝑢_𝑟+𝑑𝑟𝑑𝑠_𝑡0𝑒^{𝐴(𝑡−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠=𝑒^𝐴𝑡𝑥₀+𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹ℎ−𝐿𝑒^𝐴𝑇𝑥₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠−𝑑𝑠_𝑇0𝑡0𝑒^{𝐴(𝑡−𝑠)}𝐵𝐵^∗𝑒^{𝐴∗(𝑡−𝑠)}𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿𝑒^{𝐴(𝑇−𝑟)}×𝑓𝑟,𝑥_𝑟,𝑢_𝑟𝑑𝑠𝑑𝑟=𝑒^𝐴𝑡𝑥₀+𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹ℎ−𝐿𝑒^𝐴𝑇𝑥₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠−^𝑇₀𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿𝑒^{𝐴(𝑇−𝑟)}𝑓𝑟,𝑥_𝑟,𝑢_𝑟𝑑𝑟=𝑒^𝐴𝑡𝑥₀+𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹ℎ−𝐿𝑒^𝐴𝑇𝑥₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠−𝑄_𝑡𝑒^{𝐴∗(𝑇−𝑡)}𝐿^∗𝑄_𝑇⁻¹𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠. (32) According to Lemma 6, there is a unique pair (𝑥,𝑢)∈𝑋×𝑈_ad, satisfying (31) and (32). So, 𝑢∈𝑈_ad. Furthermore, we have 𝐿𝑥_𝑇=𝐿𝑒^𝐴𝑇𝑥₀+𝑄_𝑇𝐿^∗𝑄_𝑇⁻¹ℎ−𝐿𝑒^𝐴𝑇𝑥₀+^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠−𝑄_𝑇𝐿^∗𝑄_𝑇⁻¹𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠=𝐿𝑒^𝐴𝑇𝑥₀+𝐿𝑄_𝑇𝐿^∗𝑄_𝑇⁻¹ℎ−𝐿𝑒^𝐴𝑇𝑥₀+𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠−𝐿𝑄_𝑇𝐿^∗𝑄_𝑇⁻¹𝐿^𝑇₀𝑒^{𝐴(𝑇−𝑠)}𝑓𝑠,𝑥_𝑠,𝑢_𝑠𝑑𝑠=ℎ. (33) Thus, there is 𝑢∈𝑈_ad which steers 𝑥₀ to 𝑥_𝑇 with 𝐿𝑥_𝑇=ℎ. This means that the semilinear system (1) is 𝐿-partially complete controllable on [0,𝑇] as desired.

Remark 8. Decomposing 𝑄_𝑇 in the form 𝑄_𝑇=𝑄_𝑇𝑅_𝑇𝑅^∗_𝑅𝑃_𝑇, (34) where 𝑅_𝑇∶𝐻^⟂→𝐻 and 𝑃_𝑇∶𝐻^⟂→𝐻^⟂ are other components of 𝑄_𝑇 besides 𝑄_𝑇 and 𝐻^⟂ is an orthogonal complement of 𝐻 in 𝑋, one can calculate ⟨𝑄_𝑇ℎ,ℎ⟩=⟨𝑄_𝑇ℎ₁,ℎ₁⟩+2⟨𝑅_𝑇ℎ₂,ℎ₁⟩+⟨𝑃_𝑇ℎ₂,ℎ₂⟩, (35) where ℎ₁=𝐿ℎ∈𝐻 and ℎ₂=ℎ−𝐿ℎ∈𝐻^⟂. Therefore, the coercivity of 𝑄_𝑇 implies the same of𝑄_𝑇. But, the converse is not true. Theorem 7 is powerful in the cases when 𝑄_𝑇 is coercive, but 𝑃_𝑇 is not.

Example 9. Theorem 7 establishes just sufficient condition of 𝐿-partial complete controllability. In this example we will demonstrate that this is not a necessary condition. We will consider a simple case of 𝐿=𝐼 when 𝐿-partial complete controllability reduces to complete controllability. Consider the one-dimensional control system 𝑥^′_𝑡=2𝑥_𝑡+2𝑢_𝑡,𝑥₀∈ℝ. (36) This is a linear system, and the controllability operator of this system is equal to ^𝑇₀4𝑒^4𝑡𝑑𝑡=𝑒^4𝑇−1>0forevery𝑇>0. (37) According to the theory of controllability for linear systems, this system is controllable (completely) for every 𝑇>0.
Have another look at this system by writing it as 𝑥^′_𝑡=𝑥_𝑡+𝑢_𝑡+𝑓𝑥_𝑡,𝑢_𝑡,𝑥₀∈ℝ, (38) where 𝑓(𝑥,𝑢)=𝑥+𝑢. (39) Here, 𝑓 satisfies the Lipschitz condition with 𝐾=1. Also, 𝐴=𝐵=1, implying ‖𝐵‖=1 and 𝑀=sup_[0,𝑇]‖𝑒^𝐴𝑡‖=𝑒^𝑇. Furthermore, 𝑄_𝑇=^𝑇₀𝑒^2𝑡𝑒𝑑𝑡=^2𝑇−12. (40) So, ‖𝑄_𝑇‖=𝛾=(𝑒^2𝑇−1)/2. Then the inequality (30) becomes 1+𝑒^𝑇+2𝑒^𝑇𝑒^𝑇𝑒−1^𝑇𝑇<1. (41) The limit of the left-hand side in this inequality when 𝑇→∞ is equal to ∞. This means that there is a sufficiently large 𝑇 such that the conditions of Theorem 7 do not hold for this 𝑇, although the system under consideration is completely controllable. Thus, Theorem 7 states a sufficient condition which is not a necessary condition.

4. Examples

We demonstrate the features of 𝐿-partial complete controllability in the following examples of control systems.

Example 1. Consider the system of differential equations 𝑥^′_𝑡=𝑦_𝑡+𝑏𝑢_𝑡,𝑥₀𝑦∈ℝ,_′𝑡=𝑓𝑡,𝑥_𝑡,𝑦_𝑡,𝑢_𝑡,𝑦₀∈ℝ (42) on [0,𝑇], where 𝑢∈𝑈_ad=𝐶(0,𝑇;ℝ). Besides the complete controllability property, that is, 𝑥,𝑦∈ℝ²∶∃𝑢∈𝑈_adsuchthat𝑥_𝑇,𝑦_𝑇=𝑥,𝑦=ℝ², (43) we can investigate the partial complete controllability property, that is, 𝑥∈ℝ∶∃𝑢∈𝑈_adsuchthat𝑥_𝑇=𝑥=ℝ. (44) We can write this system in ℝ² as the following semilinear system: 𝑧^′_𝑡=𝐴𝑧_𝑡+𝐹𝑡,𝑧_𝑡,𝑢_𝑡+𝐵𝑢_𝑡, (45) where 𝑧_𝑡=𝑥_𝑡𝑦_𝑡𝑏0,𝐹()=0𝑓,,𝐴=0100𝐵=𝑡,𝑧,𝑢𝑡,𝑥,𝑦,𝑢 (46) assuming that 𝑥𝑦𝑧=. (47) It can be calculated that 𝑒^𝐴𝑡==+1𝑡0110010𝑡00. (48) Hence, 𝑒^𝐴𝑡≤1+𝑡≤1+𝑇,0≤𝑡≤𝑇. (49) The controllability operator is 𝑄_𝑇=^𝑇₀𝑒^𝐴𝑡𝐵𝐵^∗𝑒^𝐴∗𝑡𝑑𝑡=𝑏²𝑇1000. (50) Hence, 𝑄_𝑇 is not coercive, and the conditions for complete controllability, based on coercivity of 𝑄_𝑇, fail for this example. Although system (42) can still be complete controllable for properly selected functions 𝑓, we can investigate the partial complete controllability for this system being interested in just the first component 𝑥_𝑡 of 𝑧_𝑡.
Let 𝐿=[10]. Then 𝑄_𝑇=𝐿𝑄_𝑇𝐿^∗=𝑏²𝑇>0. (51) This means that the linear system associated with the semilinear system (45) is 𝐿-partially complete controllable. Furthermore, the inequality (30) becomes 𝑏1+²𝑇(1+𝑇)𝑏²𝑇+𝑏(1+𝑇)𝑏²𝑇(1+𝑇)𝑇𝐾<1, (52) or, simplifying, 𝑏𝐾<(1+𝑇)1+𝑇+2𝑏𝑇+𝑏𝑇². (53) This establishes a relation between Lipschitz coefficient 𝐾 and terminal time moment 𝑇. Depending on 𝐾, 𝑇 must be taken sufficiently large to satisfy (53). So, the system (42) is 𝐿-partially complete controllable for the time 𝑇 if the Lipschitz coefficient 𝐾, related to 𝑓, satisfies (53).

Example 2. Delay equations are typical for application of partial controllability concepts. Consider a nonlinear delay equation 𝑥^′_𝑡=𝑎𝑥_𝑡+𝑏𝑢_𝑡+𝑓𝑡,𝑥_𝑡,⁰_−𝜀𝑥_𝑡+𝜃𝑑𝜃,𝑢_𝑡,𝑥₀=𝜉,𝑥_𝜃=𝜂_𝜃,−𝜀≤𝜃≤0, (54) on [0,𝑇], where 0<𝜀<𝑇, 𝜂∈𝐿₂(−𝜀,0;ℝ), and 𝑢∈𝑈_ad=𝐶(0,𝑇;ℝ).
Similar to Example 3, introduce the function 𝑥∶[0,𝑇]→𝐿₂(−𝜀,0;ℝ) by 𝑥_𝑡𝜃=𝑥_𝑡+𝜃,0≤𝑡≤𝑇,−𝜀≤𝜃≤0. (55) This function satisfies 𝑥^′_𝑡=𝑑𝑑𝜃𝑥_𝑡,𝑥₀=𝜂,0<𝑡≤𝑇. (56) Denote by 𝒯_𝑡, 𝑡≥0, the semigroup generated by the differential operator 𝑑/𝑑𝜃 and let Γ be the integral operator from 𝐿₂(−𝜀,0;ℝ) to ℝ, defined by Γℎ=⁰_−𝜀ℎ_𝜃𝑑𝜃,ℎ∈𝐿₂(−𝜀,0;ℝ), (57) noticing that ‖Γ‖≤√𝜀. Then for 𝑦_𝑡=𝑥_𝑡𝑥_𝑡𝜉𝜂,𝜁=∈ℝ×𝐿₂(−𝜀,0;ℝ), (58) we can write system (54) as 𝑦^′_𝑡=𝐴𝑦_𝑡+𝐹𝑡,𝑦_𝑡,𝑢_𝑡+𝐵𝑢_𝑡,𝑦₀=𝜁, (59) where 0𝑑𝐴=𝑎0=𝑓(𝑑𝜃,𝐹𝑡,𝑦,𝑢𝑡,𝑥,Γ)0,𝑏0,𝑥,𝑢𝐵= (60) assuming that 𝑥𝑦=𝑥∈ℝ×𝐿₂(−𝜀,0;ℝ). (61) The semigroup, generated by 𝐴, has the form 𝑒^𝐴𝑡=𝑒^𝑎𝑡00𝒯_𝑡,𝑡≥0. (62) Therefore, the controllability operator for system (54) can be calculated as 𝑄_𝑇=^𝑇₀𝑒^𝐴𝑡𝐵^∗𝐵𝑒^𝐴∗𝑡𝑑𝑡=^𝑇₀𝑏²𝑒^2𝑎𝑡0=𝑏00𝑑𝑡₂𝑒^2𝑎𝑇−102𝑎.00 (63) This is definitely not a coercive operator.

Taking into account that the original system is given by (54) and (59) is just representation of (54) in the standard form, which enlarges the original state space ℝ to ℝ×𝐿₂(−𝜀,0;ℝ), we observe that the complete controllability for system (54) is in fact 𝐿-partial complete controllability for system (59) if 𝐿=[10]∶ℝ×𝐿₂(−𝜀,0;ℝ)⟶ℝ. (64) Calculating partial controllability operator, we obtain 𝑄_𝑇=𝐿𝑄_𝑇𝐿^∗=𝑏²𝑒^2𝑎𝑇−12𝑎>0, (65) which is coercive.

Furthermore, using 𝑒^𝐴𝑡≤1+𝑒^𝑎𝑇,𝑄_𝑇𝑏=𝛾=²𝑒^2𝑎𝑇−1,2𝑎 (66) we write the inequality (30) in the form 1+1+𝑒^𝑎𝑇+2𝑎𝑏𝑒^𝑎𝑇−11+𝑒^𝑎𝑇𝑇𝐾<1. (67) If the Lipschitz coefficient 𝐾 of the function 𝐹 and terminal time moment 𝑇 satisfy this inequality, then system (54) is completely controllable, which in turn means that system (59) is 𝐿-partially complete controllable.

5. Conclusion

In this paper a sufficient condition for partial complete controllability of a semilinear control system is proved. This is a continuation of the pioneering research that has been done by Bashirov et al. [20, 21] about partial controllability concepts. A research in this way, concerning partial complete and approximate controllability for semilinear deterministic and stochastic systems, has already been done and awaiting for publication. There are other kinds of systems which besides semilinearity include other features, for example, impulsiveness, fractional derivative issue, and so forth. The result of this paper can be extended to these systems as well.