2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela. Electronic Journal of Differential Equations, Conference 13, 2005, pp. 75–88. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXACT CONTROLLABILITY OF A NON-LINEAR GENERALIZED DAMPED WAVE EQUATION: APPLICATION

TO THE SINE-GORDON EQUATION

HUGO LEIVA

Abstract. In this paper, we give a sufficient conditions for the exact controllability of the non-linear generalized damped wave equation

ẅ + ηẇ + γA^{β}w = u(t) + f(t, w, u(t)),

on a Hilbert space. The distributed control u ∈ L^{2} and the operator A is positive definite self-adjoint unbounded with compact resolvent. The nonlinear term f is a continuous function on t and globally Lipschitz in the other variables. We prove that the linear system and the non-linear system are both exactly controllable; that is to say, the controllability of the linear system is preserved under the non-linear perturbation f. As an application of this result one can prove the exact controllability of the Sine-Gordon equation.

1. Introduction

In this paper, we give sufficient conditions for the exact controllability of the following non-linear generalized damped wave equation on a Hilbert space X,

ẅ + ηẇ + γA^{β}w = u(t) + f(t, w, u(t)), t ≥ 0,  (1.1)

where γ > 0, η > 0, β ≥ 0, the distributed control u is in L^{2}(0, t_{1}; X), and A : D(A) ⊂ X → X is a positive definite self-adjoint unbounded linear operator in X with compact resolvent. This implies the following spectral decomposition of the operator A:

Ax = ∑_{n=1}^{∞} λ_{n} ∑_{k=1}^{γ_{n}}⟨x, ϕ_{n,k}⟩ϕ_{n,k} = ∑_{n=1}^{∞} λ_{n}E_{n}x, x ∈ D(A).

The non-linear term f : [0, t_{1}] × X × X → X is a continuous function on t and globally Lipschitz in the other variables. i.e., there exists a constant l > 0 such that for all x_{1}, x_{2}, u_{1}, u_{2} ∈ X we have

∥f(t, x_{2}, u_{2}) − f(t, x_{1}, u_{1})∥ ≤ l {∥x_{2} − x_{1}∥ + ∥u_{2} − u_{1}∥} , t ∈ [0, t_{1}].  (1.2)

2000 Mathematics Subject Classification. 34G10, 35B40. Key words and phrases. Non-linear generalized wave equations; strongly continuous groups; exact controllability; Sine-Gordon equation. \begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos. Published May 30, 2005.

We consider the operator

A = [\begin{matrix}0 & I_{X} \\ −γA^{β} & −ηI \end{matrix}]  (1.3)

which corresponds to the equation ẅ + ηẇ + γA^{β}w = 0 written as a first order system in the space D(A^{β/2}) × X. Then we prove the following statements:

(I) A generates a strongly continuous group {T (t)}_{t∈ℝ} on D(A^{β/2}) × X such

that ∥T (t)∥ ≤ M(η, γ)e^{− \frac{η}{2} t}, t ≥ 0. (II) The linear system (1.4) (f = 0) is exactly controllable on [0, t_{1}]. (III) The non-linear system (1.1) is also exactly controllable on [0, t_{1}].

Moreover, each of the following statements are equivalent to the exact controllability of the linear system

ẅ + ηẇ + γA^{β}w = u(t) t ≥ 0,  (1.4)

(a) Each of the following finite dimensional systems is controllable on [0, t_{1}],

y′ = A_{j}P_{j}y + P_{j}Bu, y ∈ R(P_{j}); j = 1, 2, …, ∞.  (1.5)

(b) B^{∗}P_{j}^{∗}e^{A_{j}^{∗}t}y = 0, for all t ∈ [0, t_{1}], implies y = 0

(c) Rank [P_{j}B A_{j}P_{j}B A_{j}^{2}P_{j}B ⋯ A_{j}^{2γ_{j}−1}P_{j}B] = 2γ_{j}

(d) The operator W_{j}(t_{1}) : R(P_{j}) → R(P_{j}) given by

W_{j}(t_{1}) = ∫_{0}^{t_{1}} e^{−A_{j}s}BB^{∗}e^{−A_{j}^{∗}s}ds,  (1.6)

is invertible, where λ_{j} are the eigenvalues of A, {P_{j}} are the projections on the corresponding eigenspace,

B = [\begin{matrix}0 \\ I_{X} \end{matrix}] , A_{j} = [\begin{matrix}0 & 1 \\ −γλ_{j}^{β} & −η \end{matrix}] P_{j} , j ≥ 1.

The operator, W_{j}(t_{1}), allows us to compute explicitly the control u ∈ L^{2}(0, t_{1}; X) steering an initial state z_{0} to a final state z_{1} in time t_{1} > 0 for the linear system (1.4). This control is given by the formula

u(t) = B^{∗}T^{∗}(−t) ∑_{j=1}^{∞} W_{j}^{−1}(t_{1})P_{j}(T (−t_{1})z_{1} − z_{0}).  (1.7)

We use this formula to construct a sequence of controls u_{n} that converges to a control u that steers an initial state z_{0} to a final state z_{1} for the non-linear system (1.1). That is to say, we proved the exact controllability of this system.

As an application of this result we can prove the exact controllability of The Sine-Gordon Equation

\begin{matrix}w_{tt} + cw_{t} − dw_{xx} + k sin w = p(t, x), 0 < x < 1, t ∈ ℝ, \\ w(t, 0) = w(t, 1) = 0, t ∈ ℝ \end{matrix}  (1.8)

where d > 0, c > 0, k > 0 and p : ℝ×[0, 1] → ℝ is continuous and bounded function acting as an external force.

The existence of an attractor for the Sine-Gordon equation is proved in [9] where we can find a study of this equation, and the existence of bounded solutions for this model (1.8) and others similar one has been carried out recently in [5], [6] and [3]. To our knowledge, the exact controllability of this model under non-linear action of the control has not been studied before. So, in this paper we give a sufficient conditions for the exact controllability of the system (1.1) that can be applied to the following controlled Sine-Gordon equation

\begin{matrix}w_{tt} + cw_{t} − dw_{xx} + k sin w = p(t, x) + u(t, x) + g(t, w, u(t, x)), 0 < x < 1 \\ w(t, 0) = w(t, 1) = 0, t ∈ ℝ \end{matrix} (1.9)

where g : [0, t_{1}] × ℝ × ℝ → ℝ is a continuous function on t and globally Lipschitz in the other variables. i.e., there exists a constant m > 0 such that for all x_{1}, x_{2}, u_{1}, u_{2} ∈ ℝ we have

∥g(t, x_{2}, u_{2}) − g(t, x_{1}, u_{1})∥ ≤ m {∥x_{2} − x_{1}∥ + ∥u_{2} − u_{1}∥} , t ∈ [0, t_{1}]. (1.10)

This system can be written in the form of system (1.1) if we choose X = L^{2}[0, 1], Aϕ = −ϕ_{xx}, with domain D(A) = H^{2} ∩ H_{0}^{1} and f(t, w, u) = −k sin w + p(t, ·) + g(t, w, u). Moreover, the exact controllability of (1.9) does not depend on the bounded function p(t, ·).

Also, in [4] the authors study the exact null controllability of the second order linear equation

ẅ + ρA^{r}ẇ + Aw = u(t), ρ > 0, \frac{1}{2} ≤ r ≤ 1, t ≥ 0,  (1.11)

where the distributed control u ∈ L^{2}(0, t_{1}; X) and A : D(A) ⊂ X → X is a positive definite self-adjoint unbounded linear operator in X with compact resolvent. They prove that if \frac{1}{2} ≤ r < 1, then the system (1.11) is exactly null controllable on [0, t_{1}]. However, if α = 1, the system (1.11) is not exactly null controllable. In [2, Example 3.27] it is shown that exact null controllability of an infinite dimensional system does not imply exact controllability of the system.

2. Notation and Preliminaries

The fact that A : D(A) ⊂ X → X is a positive definite self-adjoint unbounded linear operator in X with compact resolvent implies the following:

(a) The spectrum of A consists of only eigenvalues

0 < λ_{1} < λ_{2} < ⋯ < λ_{n} → ∞,

Each λ_{j} has finite multiplicity, γ_{n}, equal to the dimension of the corresponding eigenspace.

(b) There exists a complete orthonormal set {ϕ_{n,k}} of eigenvectors of A.

(c) For all x ∈ D(A) we have

Ax = ∑_{n=1}^{∞} λ_{n} ∑_{k=1}^{γ_{n}}⟨x, ϕ_{n,k}⟩ϕ_{n,k} = ∑_{n=1}^{∞} λ_{n}E_{n}x,  (2.1)

where ⟨·, ·⟩ is the inner product in X and

E_{n}x = ∑_{k=1}^{γ_{n}}⟨x, ϕ_{n,k}⟩ϕ_{n,k}.  (2.2)

So, {E_{n}} is a family of complete orthogonal projections in X and x = ∑_{n=1}^{∞} E_{n}x, x ∈ X.

(d) −A generates an analytic semigroup {e^{−At}} given by

e^{−At}x = ∑_{n=1}^{∞} e^{−λ_{n}t}E_{n}x.  (2.3)

(e) The fractional powered spaces X^{r} are given by

X^{r} = D(A^{r}) = {x ∈ X : ∑_{n=1}^{∞}(λ_{n})^{2r}∥E_{n}x∥^{2} < ∞}, r ≥ 0,

with the norm

∥x∥_{r} = ∥A^{r}x∥ = { ∑_{n=1}^{∞} λ_{n}^{2r}∥E_{n}x∥^{2}}^{1/2}, x ∈ X^{r},

and

A^{r}x = ∑_{n=1}^{∞} λ_{n}^{r}E_{n}x.  (2.4)

Also, for r ≥ 0 we define Z_{r} = X^{r} × X, which is a Hilbert Space endow with the norm

∥ [\begin{matrix}w \\ v \end{matrix}] ∥_{Z_{r}}^{2} = ∥w∥_{r}^{2} + ∥v∥^{2}.

Using the change of variables w′ = v, the second order equation (1.1) can be written as a first order system of ordinary differential equations in the Hilbert space Z_{β/2} = D(A^{β/2}) × X = X^{β/2} × X as

z′ = Az + Bu + F (t, z, u(t)) z ∈ Z_{β/2}, t ≥ 0,  (2.5)

where

z = [\begin{matrix}w \\ v \end{matrix}] , B = [\begin{matrix}0 \\ I_{X} \end{matrix}] , A = [\begin{matrix}0 & I_{X} \\ −γA^{β} & −ηI_{X} \end{matrix}] .  (2.6)

is an unbounded linear operator with domain D(A) = D(A^{β}) × X and

F (t, z, u) = [\begin{matrix}0 \\ f(t, w, u) \end{matrix}] ,  (2.7)

is a function F : [0, t_{1}] × Z_{β/2} × X → Z. Since X^{β/2} is continuously included in X we obtain for all z_{1}, z_{2} ∈ Z_{β/2} and u_{1}, u_{2} ∈ X that

∥F (t, z_{2}, u_{2}) − F (t, z_{1}, u_{1})∥_{Z_{β/2}} ≤ L {∥z_{2} − z_{1}∥ + ∥u_{2} − u_{1}∥} , t ∈ [0, t_{1}]. (2.8)

In this paper, without lose of generality we shall assume the following condition

η^{2} < 4γλ_{1}^{β}.

3. The Uncontrolled Linear Equation

In this section we shall study the well-posedness of the abstract linear Cauchy initial-value problem

\begin{matrix}z′ = Az, (t ∈ ℝ) \\ z(0) = z_{0} ∈ D(A), \end{matrix}  (3.1)

which is equivalent to prove that the operator A generates a strongly continuous group. To this end, we shall use the following Lema from [7].

Lemma 3.1. Let Z be a separable Hilbert space and {A_{n}}_{n≥1}, {P_{n}}_{n≥1} two families of bounded linear operators in Z with {P_{n}}_{n≥1} being a complete family of orthogonal projections such that

A_{n}P_{n} = P_{n}A_{n}, n = 1, 2, 3, …  (3.2)

Define the family of linear operators

T (t)z = ∑_{n=1}^{∞} e^{A_{n}t}P_{n}z, t ≥ 0.  (3.3)

Then (a) T (t) is a linear bounded operator if

∥e^{A_{n}t}∥ ≤ g(t), n = 1, 2, 3, …  (3.4)

for some continuous real-valued function g(t).

(b) Under the condition (3.4) {T (t)}_{t≥0} is a C_{0}-semigroup in the Hilbert space Z whose infinitesimal generator A is given by

Az = ∑_{n=1}^{∞} A_{n}P_{n}z, z ∈ D(A)  (3.5)

with D(A) = {z ∈ Z : ∑_{n=1}^{∞} ∥A_{n}P_{n}z∥^{2} < ∞}

(c) the spectrum σ(A) of A is given by

σ(A) = \overline{⋃_{n=1}^{∞} σ(Ā_{n})},  (3.6)

where Ā_{n} = A_{n}P_{n}.

Theorem 3.2. The operator A given by (2.6), is the infinitesimal generator of a strongly continuous group {T (t)}_{tℝ} given by

T (t)z = ∑_{n=1}^{∞} e^{A_{n}t}P_{n}z, z ∈ Z_{β/2}, t ≥ 0  (3.7)

where {P_{n}}_{n≥0} is a complete family of orthogonal projections in the Hilbert space Z_{β/2}: P_{n} = diag[E_{n}, E_{n}], n ≥ 1, and

A_{n} = B_{n}P_{n}, B_{n} = [\begin{matrix}0 & 1 \\ −γλ_{n}^{β} & −η \end{matrix}] , n ≥ 1.  (3.8)

This group decays exponentially to zero. In fact, we have the estimate ∥T (t)∥ ≤ M(η, γ)e^{− \frac{η}{2} t}, t ≥ 0, where

\frac{M(η, γ)}{2\sqrt{2}} = sup_{n≥1} \{2|\frac{η ± \sqrt{4γλ_{n}^{β} − η^{2}}}{\sqrt{η^{2} − 4γλ_{n}^{β}}} |, |(2 + γ)\sqrt{\frac{λ_{n}^{β}}{4γλ_{n}^{β} − η^{2}}} |\}.

Proof. Computing Az yields,

Az = [\begin{matrix}0 & I \\ −γA^{β} & −η \end{matrix}] [\begin{matrix}w \\ v \end{matrix}]

= [\begin{matrix}v \\ −γA^{β}w − ηv \end{matrix}]

= [\begin{matrix}∑_{n=1}^{∞} E_{n}v \\ −γ ∑_{n=1}^{∞} λ_{n}^{β}E_{n}w − η ∑_{n=1}^{∞} E_{n}v \end{matrix}]

= ∑_{n=1}^{∞} [\begin{matrix}E_{n}v \\ −γλ_{n}^{β}E_{n}w − ηE_{n}v \end{matrix}]

= ∑_{n=1}^{∞} [\begin{matrix}0 & 1 \\ −γλ_{n}^{β} & −η \end{matrix}] [\begin{matrix}E_{n} & 0 \\ 0 & E_{n} \end{matrix}] [\begin{matrix}w \\ v \end{matrix}]

= ∑_{n=1}^{∞} A_{n}P_{n}z.

It is clear that A_{n}P_{n} = P_{n}A_{n}. Now, we need to check condition (3.4) from Lemma 3.1. To this end, compute the spectrum of the matrix B_{n}. The characteristic equation of B_{n} is given by

λ^{2} + ηλ + γλ_{n}^{β} = 0,

and the eigenvalues σ_{1}(n), σ_{2}(n) of the matrix B_{n} are given by

σ_{1}(n) = −c + il_{n}, σ_{2}(n) = −c − il_{n},

where,

c = \frac{η}{2} and l_{n} = \frac{1}{2}\sqrt{4γλ_{n}^{β} − η^{2}}.

Therefore,

e^{B_{n}t} = e^{−ct}\{ cos l_{n}tI + \frac{1}{l_{n}} (B_{n} + cI)\}

= e^{−ct} [\begin{matrix}cos l_{n}t + \frac{η}{2l_{n}} sin l_{n}t & \frac{sin l_{n}t}{l_{n}} \\ −γS(n)λ_{n}^{β/2} sin l_{n}t & cos l_{n}t − \frac{η}{2l_{n}} sin l_{n}t \end{matrix}] ,

From the above formulas, we obtain

e^{B_{n}t} = e^{−ct} [\begin{matrix}a(n) & \frac{b(n)}{l_{n}} \\ −γS(n)λ_{n}^{β/2}c(n) & d(n) \end{matrix}]

where

a(n) = cos l_{n}t + \frac{η}{2l_{n}} sin l_{n}t, b(n) = sin l_{n}t,

c(n) = sin l_{n}t, d(n) = cos l_{n}t − \frac{η}{2l_{n}} sin l_{n}t, S(n) = \sqrt{\frac{λ_{n}^{β}}{4γλ_{n}^{β} − η^{2}}} .

Now, consider z = (z_{1}, z_{2})^{T} ∈ Z_{β/2} such that ∥z∥_{Z_{β/2}} = 1. Then

∥z_{1}∥_{β/2}^{2} = ∑_{j=1}^{∞} λ_{j}^{β}∥E_{j}z_{1}∥^{2} ≤ 1 and ∥z_{2}∥_{X}^{2} = ∑_{j=1}^{∞} ∥E_{j}z_{2}∥^{2} ≤ 1.

Therefore, λ_{j}^{β/2}∥E_{j}z_{1}∥ ≤ 1, ∥E_{j}z_{2}∥ ≤ 1, j = 1, 2, . . .. and so,

∥e^{A_{n}t}z∥_{Z_{β/2}}^{2} = e^{−2ct}∥ [\begin{matrix}a(n)E_{n}z_{1} + \frac{b(n)}{l_{n}}E_{n}z_{2} \\ −γS(n)c(n)λn^{\frac{β}{2}} E_{n}z_{1} + d(n)E_{n}z_{2} \end{matrix}] ∥_{Z_{β/2}}^{2}

= e^{−2ct}∥a(n)E_{n}z_{1} + \frac{b(n)}{l_{n}}E_{n}z_{2}∥2_{\frac{β}{2}} + e^{−2ct}∥

− γS(n)c(n)λn^{\frac{β}{2}} E_{n}z_{1} + d(n)E_{n}z_{2}∥_{X}^{2}

= e^{−2ct} ∑_{j=1}^{∞} λ_{j}^{β}∥E_{j}(a(n)E_{n}z_{1} + \frac{b(n)}{l_{n}}E_{n}z_{2})∥^{2}

+ e^{−2ct} ∑_{j=1}^{∞} ∥E_{j}( − γS(n)c(n)λn^{\frac{β}{2}} E_{n}z_{1} + d(n)E_{n}z_{2})∥^{2}

= e^{−2ct}λ_{n}^{β}∥a(n)E_{n}z_{1} + \frac{b(n)}{l_{n}}E_{n}z_{2}∥^{2} + e^{−2ct}∥

− γS(n)c(n)λn^{\frac{β}{2}} E_{n}z_{1} + d(n)E_{n}z_{2}∥^{2}

≤ e^{−2ct}(|a(n)| + |\frac{λ^{\frac{β}{2}}}{λ_{n}^{α}} b(n)|)^{2} + e^{−2ct}(|γS(n)c(n)| + |d(n)|)^{2},

where

|\frac{λn^{\frac{β}{2}}}{l_{n}} b(n)| = |\sqrt{\frac{λ_{n}^{β}}{η^{2} − 4γλ_{n}^{β}}} |.

If we set,

\frac{M(η, γ)}{2\sqrt{2}} = sup_{n≥1} \{2|\frac{η ± \sqrt{4γλ_{n}^{β} − η^{2}}}{\sqrt{η^{2} − 4γλ_{n}^{β}}} |, |(2 + γ)\sqrt{\frac{λ_{n}^{β}}{4γλ_{n}^{β} − η^{2}}} |\},

we have,

∥e^{A_{n}t}∥ ≤ M(η, γ)e^{−ct}, t ≥ 0, n = 1, 2, ….

Hence, applying Lemma 3.1 we obtain that A generates a strongly continuous group given by (3.7). Next, we will prove this group decays exponentially to zero. In fact,

∥T (t)z∥^{2} ≤ ∑_{n=1}^{∞} ∥e^{A_{n}t}P_{n}z∥^{2}

≤ ∑_{n=1}^{∞} ∥e^{A_{n}t}∥^{2}∥P_{n}z∥^{2}

≤ M^{2}(η, γ)e^{−2ct} ∑_{n=1}^{∞} ∥P_{n}z∥^{2}

= M^{2}(η, γ)e^{−2ct}∥z∥^{2}.

Therefore, ∥T (t)∥ ≤ M(η, γ)e^{−ct}, t ≥ 0. □

4. Exact Controllability of the Linear System

Now, we shall give the definition of controllability in terms of the linear systems

z′ = Az + Bu z ∈ Z_{β/2}, t ≥ 0,  (4.1)

where

z = [\begin{matrix}w \\ v \end{matrix}] , B = [\begin{matrix}0 \\ I_{X} \end{matrix}] , A = [\begin{matrix}0 & I_{X} \\ −γA^{β} & −ηI_{X} \end{matrix}] .  (4.2)

For all z_{0} ∈ Z_{β/2} equation (4.1) has a unique mild solution given by

z(t) = T (t)z_{0} + ∫_{0}^{t} T (t − s)Bu(s)ds, 0 ≤ t ≤ t_{1}.  (4.3)

The following definition of exact controllability can be found in [2].

Definition 4.1. We say that system (4.1) is exactly controllable on [0, t_{1}], t_{1} > 0, if for all z_{0}, z_{1} ∈ Z_{β/2} there exists a control u ∈ L^{2}(0, t_{1}; X) such that the solution z(t) of (4.3) corresponding to u, satisfies z(t_{1}) = z_{1}.

Consider the bounded linear operator

G : L^{2}(0, t_{1}; U) → Z_{β/2}, Gu = ∫_{0}^{t_{1}} T (−s)B(s)u(s)ds.  (4.4)

Then the following proposition is a characterization of the exact controllability of system (4.1).

Proposition 4.2. The system (4.1) is exactly controllable on [0, t_{1}] if and only if, the operator G is surjective, that is to say

GL^{2}(0, t_{1}; X) = Range(G) = Z_{β/2}.

Now, consider the family of finite dimensional systems

y′ = A_{j}P_{j}y + P_{j}Bu, y ∈ R(P_{j}); j = 1, 2, …, ∞.  (4.5)

Then the following proposition can be shown as in [8, Lemma 1].

Proposition 4.3. The following statements are equivalent:

(a) System (4.5) is controllable on [0, t_{1}]

(b) B^{∗}P_{j}^{∗}e^{A_{j}^{∗}t}y = 0, for all t ∈ [0, t_{1}], implies y = 0

(c) Rank [P_{j}B A_{j}P_{j}B A_{j}^{2}P_{j}B ⋯ A_{j}^{2γ_{j}−1}P_{j}B] = 2γ_{j}

(d) The operator W_{j}(t_{1}) : R(P_{j}) → R(P_{j}) given by

W_{j}(t_{1}) = ∫_{0}^{t_{1}} e^{−A_{j}s}BB^{∗}e^{−A_{j}^{∗}s}ds,  (4.6)

is invertible.

Now, we are ready to formulate the main result on exact controllability of the linear system (4.1).

Theorem 4.4. The system (4.1) is exactly controllable on [0, t_{1}]. Moreover, the control u ∈ L^{2}(0, t_{1}; X) that steers an initial state z_{0} to a final state z_{1} at time t_{1} > 0 is given by the formula

u(t) = B^{∗}T^{∗}(−t) ∑_{j=1}^{∞} W_{j}^{−1}(t_{1})P_{j}(T (−t_{1})z_{1} − z_{0}).  (4.7)

Proof. Since {T (t)}_{t≥0} is a group, the operator G in (5) can be replaced by

G : L^{2}(0, t_{1}; X) → Z_{β/2}, Gu = ∫_{0}^{t_{1}} T (−s)B(s)u(s)ds.  (4.8)

Then system (4.1) is exactly controllable on [0, t_{1}] if and only if, the operator G is surjective, that is to say

GL^{2}(0, t_{1}; X) = Range(G) = Z_{β/2}.

First, we shall prove that each of the following finite dimensional systems is controllable on [0, t_{1}]

y′ = A_{j}P_{j}y + P_{j}Bu, y ∈ R(P_{j}); j = 1, 2, …, ∞.  (4.9)

In fact, we can check the condition for controllability of this systems,

B^{∗}P_{j}^{∗}e^{A_{j}^{∗}t}y = 0, ∀t ∈ [0, t_{1}], ⇒ y = 0.

In this case the operators A_{j} = B_{j}P_{j} and A are given by

B_{j} = [\begin{matrix}0 & 1 \\ −γλ_{j}^{β} & −η \end{matrix}] , A = [\begin{matrix}0 & I_{X} \\ −γA^{β} & −ηI \end{matrix}] ,

and the eigenvalues σ_{1}(j), σ_{2}(j) of the matrix B_{j} are given by σ_{1}(j) = −c + il_{j} and σ_{2}(j) = −c − il_{j}, where

c = \frac{η}{2} and l_{j} = \frac{1}{2}\sqrt{4γλ_{j}^{β} − η^{2}}.

Therefore, A_{j}^{∗} = B_{j}^{∗}P_{j} with B_{j}^{∗} = [\begin{matrix}0 & −1 \\ γλ_{j}^{β} & −η \end{matrix}] and

e^{B_{j}t} = e^{−ct}\{ cos l_{j}tI + \frac{1}{l_{j}} (B_{j} + cI)\}

= e^{−ct} [\begin{matrix}cos l_{j}t + \frac{η}{2l_{j}} sin l_{j}t & \frac{sin l_{j}t}{l_{j}} \\ −γS(j)λ_{j}^{β/2} sin l_{j}t & cos l_{j}t − \frac{η}{2l_{j}} sin l_{j}t \end{matrix}] ,

e^{B_{j}^{∗}t} = e^{−ct}\{ cos l_{j}tI + \frac{1}{l_{j}} (B_{j}^{∗} + cI)\}

= e^{−ct} [\begin{matrix}cos l_{j}t + \frac{η}{2l_{j}} sin l_{j}t & −\frac{sin l_{j}t}{l_{j}} \\ γS(j)λ_{j}^{β/2} sin l_{j}t & cos l_{j}t − \frac{η}{2l_{j}} sin l_{j}t \end{matrix}] ,

B = [\begin{matrix}0 \\ I_{X} \end{matrix}] , B^{∗} = [0, I_{X}] and BB^{∗} = [\begin{matrix}0 & 0 \\ 0 & I_{X} \end{matrix}] .

Now, let y = (y_{1}, y_{2})^{T} be in R(P_{j}) such that B^{∗}P_{j}^{∗}e^{A_{j}^{∗}t}y = 0 for all t ∈ [0, t_{1}]. Then

e^{−ct} [γS(j)λ_{j}^{β/2} sin l_{j}ty_{1} + ( cos l_{j}t − \frac{η}{2l_{j}} sin l_{j}t)y_{2}] = 0, ∀t ∈ [0, t_{1}],

which implies y = 0. From Proposition 4.3 the operator W_{j}(t_{1}) : R(P_{j}) → R(P_{j}) given by

W_{j}(t_{1}) = ∫_{0}^{t_{1}} e^{−A_{j}s}BB^{∗}e^{−A_{j}^{∗}s}ds = P_{j} ∫_{0}^{t_{1}} e^{−B_{j}s}BB^{∗}e^{−B_{j}^{∗}s}dsP_{j} = P_{j}\overline{W}_{j}(t_{1})P_{j}

is invertible. Since

∥e^{−A_{j}t}∥ ≤ M(η, γ)e^{ct}, ∥e^{−A_{j}^{∗}t}∥ ≤ M(η, γ)e^{ct},
∥e^{−A_{j}t}BB^{∗}e^{−A_{j}^{∗}t}∥ ≤ M^{2}(η, γ)∥BB^{∗}∥e^{2ct},

we have

∥W_{j}(t_{1})∥ ≤ M^{2}(η, γ)∥BB^{∗}∥e^{2ct_{1}} ≤ L(η, γ), j = 1, 2, ….

Now, we shall prove that the family of linear operators,

W_{j}^{−1}(t_{1}) = \overline{W} _{j}^{−1}(t_{1})P_{j} : Z_{β/2} → Z_{β/2}

is bounded and ∥W_{j}^{−1}(t_{1})∥ is uniformly bounded. To this end, we shall compute explicitly the matrix \overline{W} _{j}^{−1}(t_{1}). From the above formulas we obtain that

e^{B_{j}t} = e^{−ct} [\begin{matrix}a(j) & b(j) \\ −a(j) & c(j) \end{matrix}] , e^{B_{j}^{∗}t} = e^{−ct} [\begin{matrix}a(j) & −b(j) \\ d(j) & c(j) \end{matrix}] ,

where

a(j) = cos l_{j}t + \frac{η}{2l_{j}} sin l_{j}t, b(j) = \frac{sin l_{j}t}{l_{j}},

c(j) = γS(j)λ_{j}^{β/2} sin l_{j}t, d(j) = cos l_{j}t − \frac{η}{2l_{j}} sin l_{j}t, S(j) = \sqrt{\frac{λ_{j}^{β}}{4γλ_{j}^{β} − η^{2}}} .

Then

e^{−B_{j}s}BB^{∗}e^{−B_{j}^{∗}s} = [\begin{matrix}b(j)c(j)λ_{j}^{β/2}I & −b(j)d(j)I \\ −d(j)c(j)λ_{j}^{β/2}I & d^{2}(j)I \end{matrix}] .

Therefore,

\overline{W}_{j}(t_{1}) = [\begin{matrix}\frac{γS(j)λ_{j}^{β/2}}{l_{j}} k_{11}(j) & \frac{1}{l_{j}} k_{12}(j) \\ −γS(j)λ_{j}^{β/2}k_{21}(j) & k_{22}(j) \end{matrix}] ,

where

k_{11}(j) = ∫_{0}^{t_{1}} e^{2cs} sin^{2} l_{j}sds

k_{12}(j) = − ∫_{0}^{t_{1}} e^{2cs}[ sin l_{j}s cos l_{j}s − \frac{η sin^{2} l_{j}s}{2l_{j}}]ds

k_{21}(j) = ∫_{0}^{t_{1}} e^{2cs}[ sin l_{j}s cos l_{j}s − \frac{η sin^{2} l_{j}s}{2l_{j}}]ds

k_{22}(j) = ∫_{0}^{t_{1}} e^{2cs}[ cos l_{j}s − \frac{η sin l_{j}s}{2l_{j}}]^{2}ds.

The determinant Δ(j) of the matrix \overline{W}_{j}(t_{1}) is

Δ(j) = \frac{γS(j)λ_{j}^{β/2}}{l_{j}} [k_{11}(j)k_{22}(j) − k_{12}(j)k_{21}(j)]

= \frac{γS(j)λ_{j}^{β/2}}{l_{j}} \{( ∫_{0}^{t_{1}} e^{2cs} sin^{2} l_{j}sds)( ∫_{0}^{t_{1}} e^{2cs}[ cos l_{j}s − \frac{η sin l_{j}s}{2l_{j}}]^{2}ds)

− ( ∫_{0}^{t_{1}} e^{2cs}[ sin l_{j}s cos l_{j}s − \frac{η sin^{2} l_{j}s}{2l_{j}}]ds)^{2}\}.

Passing to the limit as j approaches ∞, we obtain

lim_{j→∞} Δ(j) = \frac{(e^{2ct_{1}} − 1)(1 − 2e^{ct_{1}} + e^{2ct_{1}})}{2^{4}c^{3}}.

Therefore, there exist constants R_{1}, R_{2} > 0 such that 0 < R_{1} < |Δ(j)| < R_{2}, j = 1, 2, 3, . . .. Hence,

\overline{W}^{−1}(j) = \frac{1}{Δ(j)} [\begin{matrix}k_{22}(j) & −\frac{1}{l_{j}} k_{12}(j) \\ γS(j)λ_{j}^{β/2}k_{21}(j) & \frac{γS(j)λ_{j}^{β/2}}{l_{j}} k_{11}(j) \end{matrix}] = [\begin{matrix}b_{11}(j) & b_{12}(j) \\ b_{21}(j)λ_{j}^{β/2} & b_{22}(j) \end{matrix}] ,

where b_{n,m}(j) are bounded for n = 1, 2; m = 1, 2; j = 1, 2, . . .. Using the same computation as in Theorem 3.2 we can prove the existence of a constant L_{2}(η, γ) such that

∥W_{j}^{−1}(t_{1})∥_{Z_{β/2}} ≤ L_{2}(η, γ), j = 1, 2, ….

Now, we define the linear bounded operators W (t_{1}) : Z_{β/2} → Z_{β/2}, W^{−1}(t_{1}) : Z_{β/2} → Z_{β/2}, by

W (t_{1})z = ∑_{j=1}^{∞} W_{j}(t_{1})P_{j}z, W^{−1}(t_{1})z = ∑_{j=1}^{∞} W_{j}^{−1}(t_{1})P_{j}z.

Using these definitions we see that W (t_{1})W^{−1}(t_{1})z = z and

W (t_{1})z = ∫_{0}^{t_{1}} T (−s)BB^{∗}T^{∗}(−s)zds.

Finally, we show that given z ∈ Z_{β/2} there exists a control u ∈ L^{2}(0, t_{1}; X) such that Gu = z. In fact, let u be the control

u(t) = B^{∗}T^{∗}(−t)W^{−1}(t_{1})z, t ∈ [0, t_{1}].

Then

Gu = ∫_{0}^{t_{1}} T (−s)Bu(s)ds
= ∫_{0}^{t_{1}} T (−s)BB^{∗}T^{∗}(−s)W^{−1}(t_{1})zds
= ( ∫_{0}^{t_{1}} T (−s)BB^{∗}T^{∗}(−s)ds)W^{−1}(t_{1})z
= W (t_{1})W^{−1}(t_{1})z = z.

Then the control steering an initial state z_{0} to a final state z_{1} in time t_{1} > 0 is given by

u(t) = B^{∗}T^{∗}(−t)W^{−1}(t_{1})(T (−t_{1})z_{1} − z_{0})

= B^{∗}T^{∗}(−t) ∑_{j=1}^{∞} W_{j}^{−1}(t_{1})P_{j}(T (−t_{1})z_{1} − z_{0}).

□

5. Exact Controllability of the Non-Linear System

Now, we give the definition of controllability in terms of the non-linear systems

\begin{matrix}z′ = Az + Bu + F (t, z, u(t)) z ∈ Z_{β/2}, t > 0, \\ z(0) = z_{0}. \end{matrix}  (5.1)

For all z_{0} ∈ Z_{β/2}, equation (5.1) has a unique mild solution

z(t) = T (t)z_{0} + ∫_{0}^{t} T (t)T (−s)[Bu(s) + F (s, z(s), u(s))]ds.  (5.2)

Definition 5.1. We say that system (5.1) is exactly controllable on [0, t_{1}], t_{1} > 0, if for all z_{0}, z_{1} ∈ Z_{β/2} there exists a control u ∈ L^{2}(0, t_{1}; X) such that the solution z(t) of (5.2) corresponding to u, verifies: z(t_{1}) = z_{1}.

Consider the non-linear operator G_{F} : L^{2}(0, t_{1}; U) → Z_{β/2}, given by

G_{F} u = ∫_{0}^{t_{1}} T (−s)B(s)u(s)ds + ∫_{0}^{t_{1}} T (−s)F (s, z(s), u(s))ds,  (5.3)

where z(t) = z(t; z_{0}, u) is the corresponding solution of (5.2). Then the following proposition is a characterization of the exact controllability of the non-linear system (5.1).

Proposition 5.2. The system (5.1) is exactly controllable on [0, t_{1}] if and only if, the operator G_{F} is surjective, that is to say

G_{F} L^{2}(0, t_{1}; X) = Range(G_{F}) = Z_{β/2}.

Lemma 5.3. Let u_{1}, u_{2} ∈ L^{2}(0, t_{1}; X), z_{0} ∈ Z_{β/2} and z_{1}(t; z_{0}, u_{1}), z_{2}(t; z_{0}, u_{2}) the corresponding solutions of (5.2). Then

∥z_{1}(t) − z_{2}(t)∥_{Z_{β/2}} ≤ M[∥B∥ + L]e^{MLt_{1}} \sqrt{t_{1}}∥u_{1} − u_{2}∥_{L^{2}(0,t_{1};X)},  (5.4)

where 0 ≤ t ≤ t_{1} and

M = sup_{0≤s≤t≤t_{1}}{∥T (t)∥∥T (−s)∥}.  (5.5)

Proof. Let z_{1}, z_{2} be solutions of (5.2) corresponding to u_{1}, u_{2} respectively. Then

∥z_{1}(t) − z_{2}(t)∥ ≤ ∫_{0}^{t} ∥T (t)∥∥T (−s)∥∥B∥∥u_{1}(s) − u_{2}(s)∥

+ ∫_{0}^{t} ∥T (t)∥∥T (−s)∥∥F (s, z_{1}(s), u_{1}(s)) − F (s, z_{2}(s), u_{2}(s))∥ds

≤ M[∥B∥ + L] ∫_{0}^{t} ∥u_{1}(s) − u_{2}(s)∥ + ML ∫_{0}^{t} ∥z_{1}(s) − z_{2}(s)∥ds
≤ M[∥B∥ + L]\sqrt{t_{1}}∥u_{1} − u_{2}∥ + ML ∫_{0}^{t_{1}} ∥z_{1}(s) − z_{2}(s)∥ds.

Using Gronwall’s inequality, we obtain

∥z_{1}(t) − z_{2}(t)∥_{Z_{β/2}} ≤ M[∥B∥ + L]e^{MLt_{1}} \sqrt{t_{1}}∥u_{1} − u_{2}∥_{L^{2}(0,t_{1};X)},

for 0 ≤ t ≤ t_{1}. □

Now, we are ready to formulate and prove the main Theorem of this section.

Theorem 5.4. If in addition of condition (2.8),

∥B∥ML∥W^{−1}(t_{1})∥K(t_{1})t_{1} < 1,  (5.6)

where K(t_{1}) = M[∥B∥ + L]e^{MLt_{1}}t_{1} + 1, then the non-linear system (5.1) is exactly controllable on [0, t_{1}].

Proof. Given the initial state z_{0} and the final state z_{1}, and u_{1} ∈ L^{2}(0, t_{1}; X), there exists u_{2} ∈ L^{2}(0, t_{1}; X) such that

0 = z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z_{1}(s), u_{1}(s))ds − ∫_{0}^{t_{1}} T (−s)Bu_{2}(s)ds,

where z_{1}(t) = z(t; z_{0}, u_{1}) is the corresponding solution of (5.2). Moreover, u_{2} can be chosen as

u_{2}(t) = B^{∗}T^{∗}(−t)W^{−1}(t_{1})(z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z_{1}(s), u_{1}(s))ds).

For such u_{2} there exists u_{3} ∈ L^{2}(0, t_{1}; X) such that

0 = z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z_{2}(s), u_{2}(s))ds − ∫_{0}^{t_{1}} T (−s)Bu_{3}(s)ds,

where z_{2}(t) = z(t; z_{0}, u_{2}) is the corresponding solution of (5.2), and u_{3} can be taken as follows:

u_{3}(t) = B^{∗}T^{∗}(−t)W^{−1}(t_{1})(z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z_{2}(s), u_{2}(s))ds).

Following this process we obtain two sequences

{u_{n}} ⊂ L^{2}(0, t_{1}; X), {z_{n}} ⊂ L^{2}(0, t_{1}; Z_{β/2}), (z_{n}(t) = z(t; z_{0}, u_{n})) n = 1, 2, …,

such that

u_{n+1}(t) = B^{∗}T^{∗}(−t)W^{−1}(t_{1})(z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z_{n}(s), u_{n}(s))ds)  (5.7)
0 = z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z_{n}(s), u_{n}(s))ds − ∫_{0}^{t_{1}} T (−s)Bu_{n+1}(s)ds.  (5.8)

Now, we shall prove that {z_{n}} is a Cauchy sequence in L^{2}(0, t_{1}; Z_{β/2}). In fact, from formula (5.7) we obtain that

u_{n+1}(t) − u_{n}(t)

= B^{∗}T^{∗}(−t)W^{−1}(t_{1})( ∫_{0}^{t_{1}} T (−s)(F (s, z_{n−1}(s), u_{n−1}(s)) − F (s, z_{n}(s), u_{n}(s)))ds).

Hence, using lemma 5.3 we obtain

∥u_{n+1}(t) − u_{n}(t)∥
≤ ∥B∥ML∥W^{−1}(t_{1})∥ ∫_{0}^{t_{1}} (∥z_{n}(s) − z_{n−1}(s)∥ + ∥u_{n}(s) − u_{n−1}(s)∥) ds
≤ ∥B∥ML∥W^{−1}(t_{1})∥ ∫_{0}^{t_{1}} M[∥B∥ + L]e^{MLt_{1}} \sqrt{t_{1}}∥u_{n}(s) − u_{n−1}(s)∥ds
+ ∥B∥ML∥W^{−1}(t_{1}) ∫_{0}^{t_{1}} ∥u_{n}(s) − u_{n−1}(s)∥ds.

Using Hóder’s inequality, we obtain

∥u_{n+1} − u_{n}∥_{L^{2}(0,t_{1};X)} ≤ ∥B∥ML∥W^{−1}(t_{1})∥K(t_{1})t_{1}∥u_{n+1} − u_{n}∥_{L^{2}(0,t_{1};X)}. (5.9)

Since ∥B∥ML∥W^{−1}(t_{1})∥K(t_{1})t_{1} < 1, it follows that {u_{n}} is a Cauchy sequence in L^{2}(0, t_{1}; X). Therefore, there exists u ∈ L^{2}(0, t_{1}; X) such that lim_{n→∞} u_{n} = u in L^{2}(0, t_{1}; X).

Let z(t) = z(t; z_{0}, u) the corresponding solution of (5.2). Then we shall prove that

lim_{n→∞} ∫_{0}^{t_{1}} T (−s)F (s, z_{n}(s), u_{n}(s))ds = ∫_{0}^{t_{1}} T (−s)F (s, z(s), u(s))ds.

In fact, using lemma 5.3 we obtain that

∥ ∫_{0}^{t_{1}} T (−s)[F (s, z_{n}(s), u_{n}(s)) − F (s, z(s), u(s))]ds∥
≤ ∫_{0}^{t_{1}} ML[∥z_{n}(s) − z(s)∥ + ∥u_{n}(s) − u(s)∥]ds

≤ ∫_{0}^{t_{1}} ML[M[∥B∥ + L]e^{MLt_{1}} \sqrt{t_{1}}∥u_{n} − u∥_{L^{2}(0,t_{1};X)} + ∥u_{n}(s) − u(s)∥]ds
≤ MLK(t_{1})\sqrt{t_{1}}∥u_{n} − u∥_{L^{2}(0,t_{1};X)}.

From here we obtain the result. Finally, passing to the limit in (5.8) as n approaches ∞, we obtain

0 = z_{1} − ∫_{0}^{t_{1}} T (−s)F (s, z(s), u(s))ds − ∫_{0}^{t_{1}} T (−s)Bu(s)ds.

i.e., G_{F} u = z_{1}. □

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Department of Mathematics, Universidad de los Andes, Merida 5101, Venezuela E-mail address: hleiva@ula.ve