On Controllability of Linear Systems 1

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1 On Controllability of Linear Systems 1 M.T.Nair Department of Mathematics, IIT Madras Abstract In this article we discuss some issues related to the observability and controllability of linear systems. Mainly we consider the system in the setting of finite dimensional Euclidean spaces. We do make some remarks about the applicability of the methods in the setting of infinite dimensional Hilbert space as well. The content of this article is essentially taken from project report (cf. [2]) based on a series of lectures the author had given in the Department of Mathematics, IIT Bombay, in For the lectures I had essentially used the book by Russel [6]. For the expositions on infinite dimensional controllability in this article, I referred some portions from book by Pazy [5] and the recent book by Curtain and Zwart [1] were used. Key-Words: Observability, Controllability, Linear System, Transition matrix, Controllability Grammian. CONTENTS 1. Solution of a Linear System (a) Solution of homogeneous system (b) Solution of non-homogeneous system using transition matrix 2. Controllability: In Finite Dimensional Setting (a) Definition of controllability (b) Control operator and reachable set (c) Kalman s condition for controllability (d) Controllability Grammian and steering operator 3. Controllability: In Infinite Dimensional Setting (a) Some basics of semigroup theory (b) Mild solution and controllability (c) Least-square control (d) Approximate controllability 4. Observability 1 Lectures at IIST Trivandrum on Novemebr 28-29, 212, under the auspices of NPDE-TCA 1

2 1 Solution of a Linear System Consider a linear system ẋ(t) = A(t)x(t) + f(t), (1.1) where A(t) R n n, f(t) R m and x(t) R n for each t I := (a, b). THEOREM 1.1. (Existence and Uniqueness) If A( ) is continuous and f is locally integrable on I, then for every (, x ) I R n, there exists a unique absolutely continuous function x such that (1.1) is satisfied a.e. with x( ) = x. Proof. (Sketch) Existence: For each closed and bounded interval J I, use Picard s-type iterations to obtain x L 2 (J, R n ) satisfying x(t) = x + [A(s)x(s) + f(s)]ds, t J, and use fundamental theorem of Lebesgue integration to see that ẋ(t) = A(t)x(t) + f(t), x( ) = x. Uniqueness: First observe that ẏ(t) = A(t)y(t), ż(t) = A(t) z(t) = y(t), z(t) constant. Suppose x and x are solutions, and let y = x x. For τ J, let z as above with z(τ) = y(τ). Then we have x(τ) x(τ) 2 = y(τ), z(τ) = y( ), z( ) =. This is true for all τ J sothat x = y. Then the solution can be extended to all of I, by using the fact that I is a union of an increasing sequence of closed and bounded intervals. Throughout this article, we shall assume the conditions on A( ) and f as prescribed in the above theorem. In applications the forcing function f may be of the form f(t) = F (t, u(t)), t I, for some control function u belonging to some admissible set Ũ of functions defined on I so that by choosing it appropriately the state function would behave in a specified manner. In engineering and science applications, the function u also represent certain intrinsic property of the system - such as density, specific heat etc. Another issue of practical interest would be the following: Suppose we do not know the initial state x ; but we may be knowing an observation on x, namely, an output function w based on the state function x. Then the question of interest is: 2

3 From the knowledge of the out function w can we determine the state function x uniquely? We shall see that both the above issues are related to each other. First, let us recall some preliminaries on ODE. 1.1 Solution of homogeneous system Consider the system ẋ(t) = A(t)x(t), t (a, b). (1.2) THEOREM 1.2. The following hold: (i) There exists solutions ϕ 1,..., ϕ n for (1.2) such that they are linearly independent. (ii) For every s I, ϕ 1 (s),..., ϕ n (s) are linearly independent vectors in R n. (iii) Every solution of (1.2) is a linear combination of ϕ 1,..., ϕ n. Proof. (i) Let {e 1,..., e n } be the standard basis for R n. Then, by Theorem 1.1, for each j {1,..., n}, there exists a unique solution ϕ j for (1.2) such that ϕ j ( ) = e j. Since rank[ϕ 1 ( ) ϕ 2 ( ) ϕ n ( )] = rank[e 1 e 2 e n ] = n, it can be shown that ϕ 1,, ϕ n are linearly independent. (ii) Let ϕ 1,, ϕ n be as in (i), and let s (a, b). We show that ϕ 1 (s),, ϕ n (s) are linearly independent. Assume, for a moment, that ϕ 1 (s),, ϕ n (s) are linearly dependent or some s I. Then there exists non-zero (α 1,..., α n ) R n such that n i=1 α iϕ i (s) =. Note that ϕ := n i=1 α iϕ i is a solution of (1.2) which satisfies ϕ(t) = A(t)ϕ(t), ϕ(s) =. Hence, by Theorem 1.1, ϕ is the zero function. Consequently, ϕ 1,..., ϕ n are linearly dependent; a contradiction. Thus, we have shown that ϕ 1 (s),, ϕ n (s) are linearly independent. (ii) Suppose that ψ is any solution of (1.2) and x = ψ( ) for some (a, b). By (ii), ϕ 1 ( ),..., ϕ n ( ) are linearly independent in R n, and hence they form a a basis of R n. Let α 1,..., α n be in R such that ψ( ) = α 1 ϕ 1 ( ) + + α n ϕ n ( ). Note that ϕ := n i=1 α iϕ i is a solution of (1.2) which satisfies ϕ( ) = ψ( ). Again, by Theorem 1.1, ψ(t) = ϕ(t) for all t (a, b). Thus, ψ is a linear combination of ϕ 1,..., ϕ n. Thus, we have shown that {ϕ 1,..., ϕ n } is a basis for mathcalx. By the above theorem, the set of all solutions of (1.2), i.e., X := {x C 1 (a, b) : x is a solution of (1.2)} 3

4 is a vector space over R with dim(x ) = n. Consider the matrix Φ(t) := [ϕ 1 (t),..., ϕ n (t)], t I where ϕ 1,..., ϕ n are as in Theorem 1.2. By Theorem 1.2, Φ(t) is non-singular for every t I. Also, we hve Φ(t) = A(t)Φ(t). Definition 1.3. The matrix Φ(t) is called a fundamental matrix for the system (1.2). 1.2 Solution of non-homogeneous system using transition matrix Let Φ(t) be a fundamental matrix for the system (1.2). Abusing the notation, let us denote Φ(t, s) := Φ(t)Φ(s) 1 THEOREM 1.4. The function x(t) := Φ(t, )x is the solution of (1.2) satisfying x( ) = x. Proof. Clearly, and x( ) := Φ(, )x = x. ẋ(t) = Φ(t, )x = Φ(t)Φ( ) 1 x = A(t)Φ(t)Φ( ) 1 x = A(t)x(t) Definition 1.5. The matrix Φ(t, ) is called the transition matrix for (1.2). Observe that Φ(t, t) = I, Φ(t 1, t 2 )Φ(t 2, t 3 ) = Φ(t 1, t 3 ), d dtφ(t, s) = A(t)Φ(t, s). We may observe that if A(t) is a constant matrix, say A, then Φ(t, s) := e (t s)a. Remark 1.6. For a matrix A R n n and α R n, we define e A α = k= A k α. k! Of course one has to show the convergence of the above series: 4

5 Note that, if A = [a ij ], then so that Hence, Thus, This implies that (Aα) i (Aα) i = n a ij α j j=1 n ( n a ij α j a ij 2) 1/2( n α j 2) 1/2 ( n = a ij 2) 1/2 α. j=1 Aα 2 = j=1 i=1 Hence, for every m, l with l > m, l A k α k! j=1 i=1 j=1 j=1 n [ n n (Aα) i 2 a ij 2] α 2. [ n n Aα M α, M := a ij 2] 1/2. k=m i=1 j=1 A k α M k α k N. l k=m A k α k! k= ( l k=m M k ) α. k! Thus, the series k= Ak α k! is convergent in R n, and hence the linear transformation e A : R n R n is well defined as ( N ) e A A k α := lim α. N k! THEOREM 1.7. The function is the solution of (1.1) with x( ) = x. t x(t) := Φ(t, )x + Φ(t, s)f(s)ds Proof. Let x(t) be as in the theorem. Since Φ(t, s) = Φ(t, )Φ(, s), we have Hence, t x(t) = Φ(t, )x + Φ(t, ) Φ(, s)f(s)ds. t ẋ(t) = Φ(t, )x + Φ(t, ) Φ(, s)f(s)ds + Φ(t, )Φ(, t)f(t) = A(t)Φ(t, )x + A(t)Φ(t, ) = A(t)x(t) + f(t), a.e.. t Φ(, s)fu(s)ds + f(t) 5

6 Thus, is the solution of (1.1) with x( ) = x. t x(t) := Φ(t, )x + Φ(t, s)f(s)ds 2 Controllability: In Finite Dimensional Setting 2.1 Definition of controllability As pointed out in the last section, in applications the forcing function f(t) may be of the form f(t) = F (t, u(t)), t I for some control function u(t) R m belonging to some admissible set Ũ of functions. We shall deal only the case when u F (t, u) is linear, i.e., there exists B(t) R n m such that F (t, u(t)) = B(t)u(t), t I. Also, we shall take Ũ to be the space L2 := L 2 (J, R m ), where J = [, t 1 ]. 2.2 Control operator and reachable set Definition 2.1. The system ẋ(t) = A(t)x(t) + B(t)u(t) (2.3) is said to be controllable over J := [, t 1 ] if for every x, x 1 R n, there exists u L 2 (J, R m ) such that x( ) = x, x(t 1 ) = x 1, and in that case u is called a control. By Theorem 1.7, the solution x(t) of (2.3) with x( ) = x is given by t x(t) := Φ(t, )x + Φ(t, s)f(s)ds For a give, t 1 I, let us define an operator from C : L 2 (J, R m ) R n by Cu := Φ(t 1, s)b(s)u(s)ds. THEOREM 2.2. The system (2.3) is controllable if and only if the operator C is onto, and in that case any u L 2 (J, R m ) such that Cu = x 1 Φ(t 1, )x is a control. 6

7 Proof. Recall that for a give u L 2 (J, R m ), the solution of ẋ(t) = A(t)x(t) + B(t)u(t) with x( ) = x is given by t x(t) := Φ(t, )x + Φ(t, s)b(s)u(s)ds. Hence, (2.3) is controllable if and only if there exists u L 2 (J, R m ) such that Cu = x 1 Φ(t 1, )x. In view of the above theorem, we have the following definition. Definition 2.3. The operator C defined above is called the control operator, and the set R(C), the range of C, is called the reachable set corresponding to the system (2.3). We have already observed that if A(t) is a constant matrix A, then Φ(t, s) = e (t s)a. Thus, if A(t) and B(t) are constant matrices A and B, respectively, then the system (2.3) takes the form and the operator C is given by Cu := ẋ(t) = Ax(t) + Bu(t) Example 2.4. (i) Consider the differential equation e (t1 s)a Bu(s)ds. ẋ(t) = x(t) + u(t). Then we know that the the solution x(t) satisfying x( ) = x is given by t x(t) = x e t t + e t s u(s)ds. Thus, the control problem is to find a function u such that That is to find u such that x 1 := x(t 1 ) = x e t1 t + e t1 s u(s)ds. e s u(s)ds = e t1 [x 1 x e t1 t ]. Note that F : u t 1 e s u(s)ds is a nonzero linear functional on L 2 (J, R), and hence it is surjective. Hence, there exists u L 2 (J, R) such that F (u) = e t1 [x 1 x e t1 t ]. (ii) Consider the system ẋ(t) = Ax(t) + Bu(t), [ ] [ ] 1 1 where A = and B =. Equivalently, 1 ẋ 1 (t) = x 1 (t) + u ẋ 2 (t) = x 2 (t). 7

8 In this case the system is not controllable: [ Suppose that the requirement on u is such that x( ) = [ ] and β 1 β 2 α 1 α 2 ] and x(t 1 ) = [ β 1 β 2 ] for some given. We know that there exists u such that x 1 ( ) = α 1 and x 1 (t 1 ) = β 1, and x 2 (t) = α 2 e t t. Hence, x 2 (t 1 ) = α 2 e t1 t. This quantity need not be equal to β 2. [ α 1 α 2 ] 2.3 Kalman s condition for controllability The above examples shows the necessity of prescribing conditions on (A, B) so that the system is controllable. We prescribe one such condition now. THEOREM 2.5. (Kalman s condition) Suppose A(t) and B(t) are constant matrices A and B, respectively. Then the system (2.3) is controllable if and only if rank[b AB A 2 B A n 1 B] = n. Proof. Let K = [B AB A 2 B A n 1 B]. Then rank(k) = n if and only if K : R mn R n is onto, if and only if K is one-one; i.e., if and only if, for y R n, K y = implies y =, i.e., if and only if y K = implies y =. Now, suppose that C is not onto, i.e., R(C) {}. Then there exists nonzero β R n such that Cu, β R n = for all u L 2 (J, R m ) i.e., In particular, taking u(s) = B e (t1 s)a β, we get e (t1 s)a Bu(s), β R nds = u L 2 (J, R m ). i.e., e (t1 s)a BB e (t1 s)a β, β R nds = u L 2 (J, R m ), B e (t1 s)a β, B e (t1 s)a β R nds = u L 2 (J, R m ), i.e., the function z(s) = B e (t1 s)a β satisfies z, z L 2 =. Hence z =, i.e., β e (t1 s)a B =. Since e (t1 s)a (t 1 s) k = I + A k, k! k=1 ( ) we have Thus, β e (t1 s)a B = β (t 1 s) k B + β A k B. k! k=1 β e (t1 s)a B = = y A k B = k N = β K = 8

9 so that K α, β = β K α = α R mn. Thus, K is not onto. Conversely, suppose K is not onto. Then there exists β such that y K =. Hence, using Cayley-Hamilton theorem and ( ), we obtain y K = = y A k B = k N = y e (t1 s)a B =. Hence, for every u L 2 (J, R m ), y e (t1 s)a Bu(s)ds =, i.e., Cu, y = for every u L 2. Thus, C is not onto. Note that in Example 2.4, we have [[ ] [ ] [ ]] [ ] K = [B AB] = =, 1 1 so that rank(k) = 1 < 2. Example 2.6. Consider the equation of the harmonic oscillator: ÿ + y = u, [ ] [ ] [ ] y 1 i.e., ẋ = Ax + Bu, where x =, A = and B =. In this case, y 1 1 [ ] 1 K = [B AB] = 1 so that rank(k) = 2. Hence, the system is controllable. 2.4 Controllability Grammian and steering operator We observe that CC is a non-negative self-adjoint linear operator from R n to itself. THEOREM 2.7. The system (2.3) is controllable over [, t 1 ] if and only if CC is positive definite, and in that case u = C (CC ) 1 [x 1 Φ(t 1, )x ] defines a control. Further, this control is the one having least norm. Proof. Since R(C ) is closed, by Projection Theorem, for every u L 2 (J, R m ), Cu = C(u 1 + u 2 ) 9

10 with u 1 R(C ), u 2 R(C ) = N(C). Hence, Cu = Cu 1 R(CC ) u L 2 (J, R m ). Thus, R(C) = R(CC ). Since CC is a linear operator from R n to itself, we have C is onto CC is bijective CC positive defiite. Thus, (2.3) is controllable if and only if CC is bijective, and in that case u := C (CC ) 1 [x 1 Φ(t 1, )x ]. satisfies Note that Cu = CC (CC ) 1 [x 1 Φ(t 1, )x ] = x 1 Φ(t 1, )x. u R(C ) = N(C). Hence, by projection theorem for Hilbert spaces, This completes the proof. u L 2 = inf{ v L 2 : v L 2, Cv = x 1 }. Let us see how C looks: Recall that Cu := Then for every β R n, Cu, β R n = = Φ(t 1, s)b(s)u(s)ds, u L 2. = u, C β. Φ(t 1, s)b(s)u(s), β R n ds u(s), B(s) Φ(t 1, s) β R n ds Thus, and hence i.e., CC β = CC = (C β)(s) = B(s) Φ(t 1, s) β, Φ(t 1, s)b(s)b(s) Φ(t 1, s) β ds, Φ(t 1, s)b(s)b(s) Φ(t 1, s) ds. 1

11 Definition 2.8. The matrix CC = Φ(t 1, s)b(s)b(s) Φ(t 1, s) ds. is called the controllability grammian for (2.3). Example 2.9. Consider the Example 2.4: ẋ(t) = x(t) + u(t). In this case, Thus, Φ(t, s) = e t s, B = [1]. (C β)(s) = e t1 s β, β R n CC = Φ(t 1, s)b(s)b(s) Φ(t 1, s) ds = e 2(t1 s) ds, (CC ) 1 α α = e 2(t1 s) ds, α Rn u(s) = C (CC ) 1 [x 1 Φ(t 1, )x ](s) = C (CC ) 1 [x 1 e t1 t x ](s) = et1 s [x 1 e t1 t x ] e 2(t1 s) ds Of course, in this case, we obtain the control from the definition of C, since is satisfied for We observe that whereas Note that so that Cu := Φ(t 1, s)b(s)u(s)ds = u(s) 2 ds = ũ(s) 2 ds = ( t 1 ũ := x 1 e t1 t x e t1 s ds. e t1 s u(s)ds = x 1 e t1 t x e 2(t1 s) x 1 e t1 t x 2 ( t 1 = x 1 x et1 t 2 e 2(t1 s) ds) 2 t1 e 2(t1 s) ds, x 1 e t1 t x 2 ( t 1 = x 1 et1 t x 2 (t 1 ) e t1 s ds) 2 ( t 1. e t1 s ds) 2 ) 2 e t1 s ds ( e 2(t1 s) ds)(t 1 ) u(s) 2 ds ũ(s) 2 ds. 11

12 LEMMA 2.1. The function y(t) := Φ(τ, t) x τ is the solution of ẏ(t) = A(t) y(t), y(τ) = x τ. Proof. For α R n, let x(t) = Φ(t, τ)α and let y(t) = Φ(τ, t) x τ. Note that But, x τ, α = x τ, Φ(τ, t)φ(t, τ)α = Φ(τ, t) x τ, Φ(t, τ)α = y(t), x(t). d y(t), x(t) dt = ẏ(t), x(t) + y(t), ẋ(t) = ẏ(t), x(t) + y(t), A(t)x(t) = ẏ(t), x(t) + A(t) y(t), x(t) = ẏ(t) + A(t) y(t), Φ(t, τ)α = Φ(t, τ) [ẏ(t) + A(t) y(t)], α. Thus, Φ(t, τ) [ẏ(t) + A(t) y(t)], α = for every α R n so that Φ(t, τ) [ẏ(t) + A(t) y(t)] =. Since Φ(t, τ) is an invertible matrix, we have This completes the proof. ẏ(t) + A(t) y(t) =. Definition The system ẏ(t) = A(t) y(t) (2.4) is called the dual of the system ẋ(t) = A(t)x(t). THEOREM The following are equivalent: 1. The system (2.3) is controllable over [, t 1 ]. 2. There exists c > such that B(s) Φ(t 1, s) α 2 ds c α 2 α R n. 3. The columns of B(t) Φ(t 1, t) are linearly independent functions. Proof. By Theorem 2.7, The system (2.3) is controllable over J if and only if CC is positive definite. Observe that for α R n, CC α, α = Φ(t 1, s)b(s)b(s) Φ(t 1, s) α, α ds = From the above relation, we obtain the equivalence. B(s) Φ(t 1, s) α 2 ds. 12

13 Recall that the adjoint of the operator C : L 2 (J, R m ) R n, is given by Cu := Φ(t 1, s)b(s)u(s)ds, u L 2. (C β)(s) = B(s) Φ(t 1, s) β, Hence, the inequality in Theorem 2.12 can be written as i.e., i.e., the operator C is bounded below. C α)(s) 2 ds c α 2 α R n, C α 2 L 2 (J,R m ) c α 2 R n α Rn, We know that the system (2.3) is controllable over J if and only if CC is positive definite. What can we say if it is not controllable? THEOREM The operator S = (C C) 1 C is well-defined. Given x 1 R n, let x 1 := x 1 Φ(t 1, )x and let u = S x 1. Then equivalently, and Cu x 1 = inf{ Cu x 1 : v L 2 (J, R m )}; C Cu = C x 1, u = inf{ v : C Cv = C x 1 }. Proof. We observe that, since R(C ) = N(C), the operator C C is injective on R(C ). Hence, the operator S = (C C) 1 C is well-defined. Also, u := Su satisfies C Cu = C x 1. Hence, equivalently, Cu x 1 = inf{ Cu x : v L 2 (J, R m )}. Further, since u R(C ) = N(C), we have u = inf{ v : C Cu = C x 1 }. Definition The operator S := (C C) 1 C is called the steering operator for the control system (2.3). Remark Note that the steering operator S defined above is the Moore-Penrose generalized inverse of the operator C. 13

14 3 Controllability: In Infinite Dimensional Setting Let us consider the system (2.3), when the space of values of the state function and control function are infinite dimensional. We shall consider only the time-independent case. More precisely, we consider the system ẋ(t) = Ax(t) + Bu(t), x() = x (3.5) for some given x X, where A : D(A) X X is a densely defined closed linear operator defined in a Hilbert space X and B : X U is a bounded linear operator between Hilbert spaces X and U, and t J := [, τ] for some τ >. Example 3.1. Consider the PDE: w t = 2 w + v(s, t), < s < l, t >, s2 with boundary conditions: w(, t) = = w(l, t). Taking X = U = L 2 (, l), and writing x(t) = w(, t), u(t) = v(, t), [Ax(t)](s) = 2 w s 2, the above PDE can be written as ẋ(t) = Ax(t) + u(t), where D(A) = {z L 2 (, l) : z L 2 (, l), z() = = z(l)}. Definition 3.2. The system (3.5) is said to be exactly controllable if for every x τ X, there exists u L 2 (J, U) such that there exists a differentiable function x L 2 (J, X) satisfying (3.5) and the condition x(τ) = x τ. 3.1 Some basics of semigroup theory To address the controllability issues in the above setting, we may recall certain results from the theory of semi-groups (cf. Pazy [5]): Definition 3.3. Let X be a Banach space over K which is either R or C. A family T := {T (t) : t } of bounded linear operators on X is called a strongly continuous semi-group or C -semi-group if 1. T () = I, 2. T (t + s) = T (t)t (s) t, s, 14

15 3. T (t)x x as t for every x X. Example 3.4. If X = R n and A R n n, then T (t) = e ta := I + k=1 t k k! Ak defines a C -semigroup. Further, it may be observed that if A is a diagonal matrix with diagonal entries as λ 1,..., λ n, then A k e j = λ k j e j for j = 1,..., n and for evert k N. Hence, in this case, T (t)e j = e ta e j = e j + k=1 t k k! Ak e j = e tλj e j. Thus, for every α R n, and n T (t)α = e ta α = e tλj α j e j. j=1 Example 3.5. If X be a Banach space and A : X X be a bounded linear operator. As in the last example, let T (t) = e ta := I + k=1 t k k! Ak. Then {T (t) : t } defines a C -semigroup. Since A k A k for every k N, it can be shown that e ta is well-defined bounded linear operator on X. THEOREM 3.6. If {T (t) : t } is a strongly continuous semi-group on X and if then A : D X defined by D := {x X : lim t T (t)x x t exists }, T (t)x x Ax = lim, x D, t t is a closed densely defined linear operator. Further, we have the following: 1. For every x D, t T (t)x is differentiable on (, ) and 2. There exists M 1 ω such that d T (t)x = AT (t)x, t >. dt T (t) Me ωt t. 15

16 By the above theorem, for every x D, the function x(t) := T (t)x, t, is a solution of the homogeneous differential equation ẋ(t) = Ax(t), x() = x. Definition 3.7. The operator A defined in Theorem 3.6 is called the infinitesimal generator of the semi-group T. Example 3.8. Let X be a separable Hilbert space and {v n : n N} be an orthonormal basis for X. Let (λ n ) be a sequence of positive real real numbers. For t, define T (t)x = e λnt x, v n v n, x X. n=1 Then, it can be seen that {T (t) : t } is a strongly continuous semigroup and its infinitesimal generator A is defined by Ax = λ n x, v n v n, x D, where D := {x X : λ 2 n x, v n 2 < }. n=1 n=1 Any mention of semi-group would be incomplete if one does not mention Hill-Yosida theorem which states a necessary and sufficient condition for a linear operator to be the infinitesimal generator of a strongly continuous semi-group. THEOREM 3.9. (Hille-Yosida theorem) A linear operator A : D(A) X X is the infinitesimal generator of a strongly continuous semi-group {T (t)} satisfying T (t) Me ωt for some M 1 and ω if and only if A is a closed densely defined operator, the resolvent set ρ(a) contains (ω, ) and (λi A) n M (λ ω) n λ > ω, n N. In view of Theorem 3.6, for every x D, the function x : J X defined by x(t) := T (t)x, t > is a solution of ẋ(t) = Ax(t), x() = x, (3.6) where A is the infinitesimal generator of {T (t)}. In fact, it is known that the solution of the Cauchy problem (3.6) is unique. 16

17 3.2 Mild solution and controllability In the following we assume that X is a Hilbert space. Now, let A be the infinitesemial generator A of a strongly continuous semi-group {T (t) : t }. For f L 2 (J, X), consider the non-homogeneous system ẋ(t) = Ax(t) + f(t), x() = x. (3.7) THEOREM 3.1. If (3.7) has a solution x, then it is given by x(t) = T (t)x + t T (t s)f(s) ds, t J. Definition For f L 2 (J, X), the function x defined by x(t) = T (t)x + is called a mild solution of (3.7). t T (t s)f(s) ds, t J, (3.8) Facts: A mild solution need not be a solution. If f L 2 (J, X) C 1 (J, X), then every mild solution is a solution. Now, let us consider the control system for u L 2 (J, U). Define K : L 2 (J, U) X by ẋ(t) = Ax(t) + Bu(t), (3.9) Let and Ku = τ T (τ s)bu(s) ds, u L 2 (J, U). L 2 (J, U) := {u L 2 (J, U) : T ( )x + Ku is a solution of (3.9)}. K u = Ku for u L 2 (J, U). THEOREM The system (3.9) is controllable if and only if K is surjective. Although a mild solution of (3.7) need not be a solution, most often, in the infinite dimensional setting, one looks for a control function u such that x(τ) = T (τ)x + So, we introduce the following definition. τ T (τ s)bu(s) ds. 17

18 Definition The system (3.9) is said to be controllable if for every x τ X, there exists u L 2 (J, U) such that x(τ) = T (τ)x + τ T (τ s)bu(s) ds. THEOREM The system (3.9) is controllable if and only if K is surjective. THEOREM The system (3.9) is controllable if and only if there exists c > such that K x L2 (J,U) c x X x X, and in that case, u := K (KK ) 1 [x τ T (τ)x ] defines a a control. Proof. First we recall the following relations R(K) = N(K ), R(K) = N(K ) and the fact, known as closed range theorem (cf. Nair [3], Theorem 11.5, page 355), that R(K) closed R(K ) closed. Suppose the system (3.9) is controllable. Then K : L 2 (J, U) X is surjective, so that its adjoint K : X L 2 (J, U) is injective and R(K ) is closed. Hence, by bounded inverse theorem, inverse of K is a bounded operator. Therefore, there exists c > such that K x c x for all x X. Conversely, suppose there exists c > such that K x c x for all x X. Then, K is injective and R(K ) is closed. Hence, R(K) is closed and R(K) = R(K) = N(K ) = {} = X. Thus, the system (3.9) is controllable. To see the expression for a control, we observe that, for x X, K x c x KK x, x c 2 x 2. Hence, KK is injective, so that R(KK ) dense. Also, since R(K ) is closed, for every u L 2 (J, U), Ku = K(u 1 + u 2 ) with u 1 R(K ), u 2 R(K ) = N(K) so that Thus, Ku = Ku 1 R(KK ). R(KK ) = R(K) = X. Therefore, KK : X X is bijective. Thus, u is well-defined and and hence, x τ = T (τ)x + Ku. Ku = KK (KK ) 1 [x τ T (τ)x ] = x τ T (τ)x 18

19 Now, let us give an expression for K and KK Note that for every u L 2 (J, U) and x X, Hence, Ku, x = = τ τ T (τ s)bu, x ds u, B T (τ s) x ds. (K x)(s) = B T (τ s) x, s J, and KK x = τ T (τ s)bb(s) T (τ s) x ds, x X. The above operator is also called the controllability grammian for the system (3.9). Thus the condition for mild controllability in Theorem 3.15 can be written as τ 3.3 Least-square control B(s) T (τ s) x 2 ds c 2 x 2, x X. Suppose the system (3.9) is not controllable. Then the range of the control operator is not the whole of X. More over R(K) need not be closed. Then what we look for is a function u L 2 (J, U) such that where x τ := x τ T (τ)x. Ku x τ = inf Kv x τ, v L 2 (J,U) In this regard, we have the following result from Hilbert-space theory: THEOREM Let x τ := x τ T (τ)x and u L 2 (J, U). Then the following are equivalent: 1. Ku x τ = inf v L 2 (J,U) Kv x τ, 2. K Ku = K x τ, 3. x τ R(K) + R(K). Definition A function u L 2 (J, U) is called a least-square control for the system (3.9) if Ku x τ = inf Kv x τ, v L 2 (J,U) where x τ := x τ T (τ)x. THEOREM If x τ R(K) + R(K), then there exists a unique least-square control ũ which minimizes the norm, and it is given by ũ = (K K) 1 K x τ. 19

20 3.4 Approximate controllability As in last section consider the control system ẋ(t) = Ax(t) + Bu(t), x() = x (3.1) for some given x X, where A : D(A) X X is a densely defined closed linear operator defined in a Hilbert space X which is the infinitesimal generator of a strongly continuous semi-group {T (t) : t }, B : X U is a bounded linear operator between Hilbert spaces X and U, and t J := [, τ] for some τ >. We have seen that the system (3.1) is controllable if and only if K is surjective, where Ku = τ T (τ s)bu(s) ds, u L 2 (J, U). Suppose (3.1) is not controllable. Definition The system (3.1) is approximately controllable if for every ε >, there exists u ε L 2 (J, U) such that the corresponding mild solution x ε (τ) := Ku ε + T (τ)x satisfies x ε (τ) x τ < ε, i.e., if and only if R(K) is dense in X. 4 Observability We shall link controllability of (2.3) with another concept, namely, the observability of another system. Definition 4.1. The system along with an output function ẋ(t) = A(t)x(t) + f(t) (4.11) w(t) = H(t)x(t). (4.12) is said to be observable if and only if w = = x =. Note that observability is about the reconstruction of the state x(t) in a unique manner from some attribute of it in the form of an output function w(t). Since the state function x(t) in (4.11) is completely determined by the initial state x := x( ), 2

21 THEOREM 4.2. The system (4.11)-(4.13) is observable if and only if w = = x( ) =. In particular, we have the following: THEOREM 4.3. The system ẋ(t) = A(t)x(t), w(t) = H(t)x(t) (4.13) is observable if and only if the columns of H(t)Φ(t, ) are linearly independent. Proof. Recall that the solution x(t) of ẋ(t) = A(t)x(t), x( ) = x, is given by x(t) = Φ(t, )x. Hence, the observability condition given in Theorem 4.2 is satisfied if and only if columns of H(t)Φ(t, ) are linearly independent. Let us define the operator O : R n L 2 (J, R n ) by (Oα)(t) := H(t)Φ(t, )α, α R n, t J. In terms of the operator O we have the following characterization: THEOREM 4.4. The system (4.13) is observable if and only if O is injective. How do we recover the state function x from the output function? Let us compute the adjoint of O: For every α R n, v L 2 (J, R n ), we have Oα, v = H(t)Φ(t, )α, v(t) dt J = α, Φ(t, ) H(t) v(t) dt J = α, Φ(t, ) H(t) v(t)dt. Thus, O v = Φ(t, ) H(t) v(t)dt. J J THEOREM 4.5. The system The system (4.13) is observable if and only if O O is positive definite and in that case for w R(O), α := (O O) 1 O w satisfies Oα = w. 21

22 Proof. By Theorem 4.4, (4.13) is observable if and only if O is injective. But, O is injective if and only if O is surjective, and in that case O O is positive definite. For w R(O), let α := (O O) 1 O w. Then we have (O O)α = O w. But, O is injective on R(O): (O O)β = = β = = Oβ =. Hence, we have Oα = w. This completes the proof. Definition 4.6. The operator R : L 2 (J, R m defined by Rw := (O O) 1 O w is called the reconstruction operator for the system (4.13). Remark 4.7. Note that the reconstruction operator is the Moore-Penrose generalized inverse of the operator O. 5 Relation Between Controllability and Observability THEOREM 5.1. The system (2.3) is controllable over [, t 1 ] if and only if the system y(t) = A(t) y(t), w(t) = B(t) y(t) is observable. Proof. By Lemma 2.1, is the solution of Hence, y(t) = Φ(t 1, t) α y(t) = A(t) y(t), y(t 1 ) = α. w(t) = B(t) Φ(t 1, t) α. Hence, By Theorem 4.3, y(t) = A(t) y(t), w(t) = B(t) y(t) is observable if and only if the columns of B(t) Φ(t 1, t) are linearly independent functions, and by Theorem 2.12, columns of B(t) Φ(t 1, t) are linearly independent functions if and only if (2.3) is controllable over [, t 1 ]. 22

23 References [1] Ruth F. Curtain, Hans Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer Verlag, [2] M.T. Nair, Observability and Controllability of Linear Systems, Lecture Notes, IIT Bombay, 1986 (Pre-Print) [3] M.T. Nair, Functional Analysis: A First Course, Printice-Hall of India, New Delhi, 22. [4] M.T. Nair, Linear Operator Equations: Approximation and Regularization, World Scientific, Singapore, May 29. [5] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, [6] D.L. Russel, Mathematics of finte dimensional Control Systems, Marcel Decker Inc., New York,