APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS

APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES Said Hadd and Qing-Chang Zhong Dept. of Electrical Eng. & Electronics The University of Liverpool Liverpool, L69 3GJ United Kingdom Q.Zhong@liv.ac.uk

Outline Quick overview of recent activities Motivation of the research Problem statement Methodology Main results Special cases S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 2/50

Recent research activities Control engineering Power electronics: grid connection etc Renewable energy: wind power Automotive electronics: hybrid electric vehicles Control theory Robust control: J-spectral factorisation, algebraic Riccati equations etc Time-delay systems: a series of problems Infinite-dimensional systems: feedback stabilizability, controllability S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 3/50

Key publications One research monograph IEEE Trans. Automatic Control: 7 papers Automatica: 4 papers other IEEE Transactions: 3 papers S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 4/50

Current funding EPSRC: EP/C005953/1, 126k, to expire in 09/2008 EPSRC: EP/E055877/1, 88k EPSRC: one DTA studentship, 50k EPSRC and Add2 Ltd: Dorothy Hodgkin Postgraduate Award, 90k ESPRC and Nheolis, France: Dorothy Hodgkin Postgraduate Award, 90k S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 5/50

Motivation Controllability is one of the most important properties of a control system. Basically, it tells us if a system is controllable, in other words, whether a point in the state space is reachable. The most widely known formula is the Kalman rank condition Rank( B AB A n 1 B ) = n. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 6/50

Controllability: Analysis and design Normally, controllability is used to analyse whether a system is controllable. It can also be used for design. Many flexible modes to be suppressed The number of actuators to be minimised Many locations for placing actuators The question is where to put them and how many? S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 7/50

Infinite-dimensional case For finite-dimensional systems, there are different ways to define controllability. However, they all end up with the same concept. However, this is no longer true for infinite-dimensional case. Different concepts need to be used. Approximate or exact controllability in finite or infinite time etc. (Nine different concepts are defined in Stafani s book) S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 8/50

Approximate controllability (AC) For a given arbitrary ε > 0, it is possible to steer the state from the origin to the neighbourhood, with a radius ε, of all points in the state space. This is weaker than the exact controllability, where ε = 0. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 9/50

Some notation X, U, Y are Banach spaces, p > 1 is a real number and (A, D(A)) is the generator of a C 0 -semigroup (T(t)) t 0 on X. The type of T(t) is defined as ω 0 (A) := inf{t 1 log T(t) : t > 0}. The domain D(A), which is a Banach space, is endowed with the graph norm x A := x + Ax, x D(A). Denote the resolvent set of A by ρ(a) and the resolvent operator of A by R(µ,A) := (µ A) 1 for µ ρ(a). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 10/50

The completion of X with respect to the norm x 1 = R(µ,A)x for x X and some µ ρ(a) is a Banach space denoted by X 1, which is called the extrapolation space of A. For any Banach spaces E and F, denote the Banach space of all linear bounded operators from E to F by L(E,F) with L(E) := L(E, E). The history function of z : [ r, ) Z is the function z t : [ r, 0] Z defined by z t (θ) = z(t + θ) for t 0 and θ [ r, 0] with delay r > 0. For p > 1, L p ([ r, 0],Z) is the Banach space of all p- integrable functions f : [ r, 0] Z and W 1,p ([ r, 0],Z) is its associated Sobolev space. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 11/50

Problem statement To investigate the approximate controllability of d ( ) x(t) Dxt K 0 u(t) K 1 u t = B0 u(t) + B 1 u t dt +A ( ) x(t) Dx t K 0 u(t) K 1 u t + Lxt, t 0, ( ) lim x(t) Dxt K 0 u(t) K 1 u t = z0, t 0 x 0 = ϕ, u 0 = ψ. (1) Here x : [ r, ) X, u : [ r, ) U, the operators D,L : W 1,p ([ r, 0],X) X, K 1,B 1 : W 1,p ([ r, 0],U) X and B 0,K 0 : U X are linear bounded, and A is the generator of a C 0 -semigroup T := (T(t)) t 0 on X. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 12/50

How general it is? Distributed delays in the state AND the input, could be multiple discrete delays Delays in the derivative: neutral K 0 0 and K 1 0. d ( ) x(t) Dxt K 0 u(t) K 1 u t = B0 u(t) + B 1 u t dt +A ( ) ( x(t) Dx t K 0 u(t) ) K 1 u t + Lxt, t 0, lim x(t) Dxt K 0 u(t) K 1 u t = z0, t 0 x 0 = ϕ, u 0 = ψ. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 13/50

Background This area had been very active during 70s and 80s, focusing on retarded systems with state or input delays. Recent works include: 1) Diblik et al: Using a representation of solutions with the aid of a discrete matrix delayed exponential for linear discrete-time systems with state delays; 2) Sun et al: Using the matrix Lambert W function for linear (continuous-time) systems with state delays; 3) Rabah et al: Using the moment problem approach for neutral systems with distributed state delays. The closest one is the last one, where K 0 = 0, K 1 = 0, A = 0 and B 1 = 0. Also, the operator D is more general here. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 14/50

Key steps Transform the problem into a perturbed boundary control problem Solve the approximate controllability for the boundary control problem Then transfer the results back to the original problem S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 15/50

More notation Define Q X mϕ = ϕ, Q U mψ = ψ, D(Q X m) = W 1,p ([ r, 0],X), D(Q U m) = W 1,p ([ r, 0],U), and Q X ϕ = ϕ, D(Q X ) = {ϕ W 1,p ([ r, 0],X) : ϕ(0) = 0}, Q U ψ = ψ, D(Q U ) = {ψ W 1,p ([ r, 0],U) : ψ(0) = 0}. It is well known that Q X and Q U generate the left shift semigroups on L p ([ r, 0],X) and L p ([ r, 0],U), respectively. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 16/50

Denote by Q X 1 and Q U 1 the extension of Q X and Q U, respectively, and define the operators, for µ C, where β X := (µ Q X 1)e µ, β U := (µ Q U 1)e µ (2) e µ : Z L p ([ r, 0],Z), (e µ z)(θ) = e µθ z (3) is defined for µ C, z Z, and θ [ r, 0]. Denote by J Z (Z = X or U) the following continuous injection J Z : D(Q Z ) W 1,p ([ r, 0],Z). (4) S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 17/50

Transforming into a boundary control problem Introduce the space X := X L p ([ r, 0],X) L p ([ r, 0],U), and the matrix operator A L B 1 A m := 0 Q X m 0, 0 0 Q U m D(A m ) := D(A) W 1,p ([ r, 0],X) W 1,p ([ r, 0],U), S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 18/50

Moreover, define the boundary operators N, M : D(A m ) U := X U as ( ) ( ) 0 δ0 0 I D K1 N =, M =, 0 0 δ 0 0 0 0 and, for any u U, the operators Bu = ( B0 u 0 0 ), Ku = ( K 0 u u ) S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 19/50

Then the system (1) can be reformulated as ẇ(t) = A m w(t) + Bu(t), t 0, Nw(t) = Mw(t) + Ku(t). (5) by setting w(t) = (z(t),x t,u t ), t 0 with w(0) = = (z 0,ϕ,ψ), where z(t) = x(t) Dx t K 0 u(t) K 1 u t with limz(t) = z 0. Here, A m : X m X and t 0 N, M : X m U are linear operators with Banach spaces X, U and X m being a dense domain of X endowed with a norm, which is finer than the norm of X such that (X m, ) is complete. This is a boundary control problem. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 20/50

Verification of conditions Standard conditions for boundary control problems: (H1) A = A m with domain D(A) := KerN generates a C 0 -semi group (T (t)) t 0 on X, (H2) ImN = U. These assumptions show that the boundary value problem with M = 0 and B = 0 is well-posed in the sense that it can be reformulated as a well-posed open loop system on the state space X and control space U. Clearly N satisfies the condition (H2). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 21/50

In order to verify condition (H1), assume (A1) (Q X,β X,LJ X ) generates a regular linear system on L p ([ r, 0],X), X, X, (A2) (Q U,β U,B 1 J U ) generates a regular linear system on L p ([ r, 0],U), U, X. Define the operator A := A L B 1 0 Q X 0, 0 0 Q U D(A) := KerN = D(A) D(Q X ) D(Q U ). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 22/50

It satisfies A = A 0 + L with A 0 0 A 0 := 0 Q X 0, L := 0 0 Q U 0 L B 1 0 0 0 0 0 0 D(A 0 ) := KerN., D(L) := KerN. Clearly, A 0 generates a diagonal C 0 -semigroup on X. It is not difficult to see that if L and B 1 are admissible observation operators for Q X and Q U (in particular if (A1) and (A2) holds), respectively, then L is a Miyadera Voigt perturbation for A 0. Then A generates a C 0 -semigroup T := (T (t)) t 0 on X. Hence N satisfies the condition (H1). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 23/50

Admissible observation operator Definition 1. C L(D(A),Y ) is called an admissible observation operator for A if, for constants τ 0 and γ := γ(τ) > 0, τ 0 CT(t)x p dt γ p x p, x D(A), (6) The map Ψ x := CT( )x, defined on D(A), extends to a bounded operator Ψ : X L p loc (R +,Y ). For any x X we set Ψ(t)x := Ψ x on [0,t] and call (T(t), Ψ(t)) t 0 an observation system. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 24/50

Admissible control operator Definition 2. B L(U,X 1 ) is called an admissible control operator for (A, D(A)), the generator of a C 0 - semigroup (T(t)) t 0 on X, if for all t > 0 and u L p ([ r, 0],U) the control map Φ(t)u := takes values in X. t 0 T(t s)bu(s)ds (7) The pair (T, Φ) is called a control system represented by the operator B. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 25/50

Formal definition of AC Definition 3. Let (T, Φ) be the control system represented by the operator B. Define the reachability space R := t 0 RanΦ(t). Then, (A, B) is said to be approximately controllable if R is dense in X. This means that, for a given arbitrary ε > 0, it is possible to steer the state from the origin to the neighbourhood, with a radius ε, of all points in the state space. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 26/50

Further transformation It can be proved that the boundary control system (5) is equivalent to the distributed-parameter system ẇ(t) = Aw(t) + (B + BK)u(t), w(0) =, (8) for t 0, with A and B defined as Ax = A m x, D(A) = { x D(A m ) : Nx = Mx }. B := (µ A 1 )D µ, µ ρ(a). where D µ is the Dirichlet operator D µ := ( N Ker(µ Am )) 1 : U Ker(µ Am ). (9) S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 27/50

AC for the boundary control problem When the conditions (H1) and (H2) are satisfied, D µ exists and is bounded. For µ ρ(a), there is (A m A 1 ) = (µ A 1 )D µ N on D(A m ). Definition 4. Assume that (H1) (H2) are satisfied. The boundary value problem (5) is said to be approximately controllable if the open-loop system (A, B+BK) is (in the sense of Definition 3). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 28/50

Theorem Theorem 5. Assume that (H1) (H2) are satisfied. The boundary value problem (5) is approximately controllable if and only if, for µ ρ(a) ρ(a) and ϕ X, the fact that (I D µ M) 1 (D µ K + R(µ, A)B)u,ϕ = 0, for u U, (10) implies that ϕ = 0. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 29/50

Going back to the neutral system... S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 30/50

Dirichlet operator associated with N Note that ρ(a) = ρ(a). For µ ρ(a), one can see that the Dirichlet operator associated with N, as defined in (9), is given by ( D uv ) R(µ,A)(Le µ u + B 1 e µ v) ( µ = e µ u, uv ) U. e µ v (11) S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 31/50

More assumptions Let J : D(A) D(A) W 1,p ([ r, 0],X) W 1,p ([ r, 0],U) be the continuous injection, and set C := MJ L(D(A), U). Then C is an admissible observation operator for A under the following assumptions: (A3) (Q X,β X,DJ X ) generates a regular linear system on L p ([ r, 0],X), X, X with I X as an admissible feedback, (A4) (Q U,β U,K 1 J U ) generates a regular linear system on L p ([ r, 0],U), U, X. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 32/50

(µ) Define (µ) := I De µ R(µ,A)Le µ, µ ρ(a). By using (11), for µ ρ(a), we have ( ) (µ) (K1 e I U MD µ = µ + R(µ,A)B 1 e µ ). 0 I U (12) Thus, (µ) is invertible if and only if µ ρ(a) ρ(a). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 33/50

AC: the transformed system Definition 6. Assume that the conditions (A1) to (A4) are satisfied. Define R(t)u := t 0 (T I ) 1 (t τ)(bk + B)u(τ)dτ for t 0 and u L p loc (R +,U). The system (5) is said to be Υ-approximately controllable if Cl( t 0 P Υ (RanR(t))) = Υ, where Υ can be any of X, X, X L p ([ r, 0],X), L p ([ r, 0],X) and L p ([ r, 0],U), and P Υ is the projection operator from X to Υ. In particular, P X = I. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 34/50

w(t) = z(t) x t u t X-approximately controllable: the state z is approximately controllable; X L p ([ r, 0],X)-approximately controllable: states z and x t are approximately controllable; L p ([ r, 0],U)-approximately controllable: state u t is approximately controllable. Apparently, it is always L p ([ r, 0],U)- approximately controllable. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 35/50

AC: Transformed system Theorem 7. Assume that (A1) to (A4) are satisfied. Then the following conditions are equivalent: (i) the system (5) is X -approximately controllable, (ii) it is X L p ([ r, 0],X)-approximately controllable, (iii) for µ ρ(a) ρ(a), ϕ L q ([ r, 0],X) with 1 p + 1 q = 1 and x X, the fact that, for u U, (Ω(µ) + Γ(µ)Λ(µ))u,x + (eµ K 0 + e µ (µ) 1 Λ(µ))u,ϕ = 0 implies that x = 0 and ϕ = 0. Here, Γ(µ) := R(µ,A)Le µ (µ) 1, Ω(µ) := R(µ,A)(Le µ K 0 + B 1 e µ + B 0 ), Λ(µ) : = Ω(µ) + De µ K 0 + K 1 e µ. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 36/50

Outline of proof (I D µ M) 1 = I + Γ(µ) Γ(µ)D Γ(µ)K 1 e µ (µ) 1 I + e µ (µ) 1 D e µ (µ) 1 K 1. 0 0 I R(µ, A) = R(µ, A) R(µ, A)LR(µ, Q X ) R(µ, A)B 1 R(µ, Q U ) 0 R(µ, Q X ) 0. 0 0 R(µ, Q U ) D µ Ku + R(µ, A)Bu = Ω(µ)u e µ K 0 u. e µ u Hence, Ω(µ) + Γ(µ)Λ(µ) (I D µ M) 1 (D µ Ku + R(µ, A)Bu) = e µ K 0 + e µ (µ) 1 Λ(µ) u S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 37/50 e µ

State z(t) Assume that the conditions (A1) to (A4) are satisfied. Then the system (5) is X-approximately controllable iff, for µ ρ(a) ρ(a) and x X, the fact that (Ω(µ) + Γ(µ)Λ(µ))u,x = 0, for u U, implies that x = 0. This characterises the condition when z(t), partial state of system (5), can reach all points in X. In general, this is irrelevant to the approximate controllability of system (1). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 38/50

State x t Assume that the conditions (A1) to (A4) are satisfied. Then the system (5) is L p ([ r, 0],X)-approximately controllable iff, for µ ρ(a) ρ(a) and ϕ L q ([ r, 0],X) with 1 p + 1 q = 1, the fact that eµ (K 0 + (µ) 1 Λ(µ))u,ϕ = 0, for u U, (13) implies that ϕ = 0. This actually describes the approximate controllability of system (1). S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 39/50

AC: the neutral system Theorem 8. Assume that the conditions (A1) to (A4) are satisfied. Then the general neutral system (1) is approximately controllable iff, for µ ρ(a) ρ(a) and x X, the fact that (K0 + (µ) 1 Λ(µ))u,x = 0, for u U, implies that x = 0. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 40/50

Special case I: Difference equations with state and input delays Consider the difference equation x(t) = Dx t + K 0 u(t) + K 1 u t, t 0, (14) with x 0 = ϕ, u 0 = ψ. Note that the equation (14) is a special case of (1), with A = 0, L = 0,B 0 = 0, B 1 = 0. In this case, (µ) = I De µ, Γ(µ) = 0, Ω(µ) = 0, Λ(µ) = De µ K 0 + K 1 e µ. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 41/50

Define the operator Q I ϕ = ϕ, { } D(Q I ) = ϕ W 1,p ([ r, 0],X) : ϕ(0) = Dϕ. (µ) is invertible if and only if µ ρ(q I ). Assume that the conditions (A3) to (A4) are satisfied. Then the system (14) is approximately controllable iff, for µ ρ(q I ) and x X, the fact that (I Deµ ) 1 (K 0 + K 1 e µ )u,x = 0, for u U implies that x = 0. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 42/50

Special case II: Retarded delay systems with state and input delays Consider the following retarded delay system with finite-dimensional delay-free dynamics, which widely exist in engineering: ẋ(t) = Ax(t)+L r x(t r)+b 0 u(t)+b r u(t r), t 0, where x R n, A R n n, L r R n n, B 0 R n m, B r R n m and u R m. The variants of this system have been targets of most of the papers in this area. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 43/50

This system can be obtained from the system (1) by taking D = 0, K 0 = 0, K 1 = 0, L = L r δ r and B 1 = B r δ r, where δ r is the Dirac operator. These operators satisfy the conditions (A1) (A4). Then, (µ) = I e rµ R(µ,A)L r, µ ρ(a). Define ρ = {µ : (µ) is invertible}. Then for µ ρ(a) ρ, Γ(µ) = e rµ R(µ,A)L r (µ) 1 = (µ) 1 I, Λ(µ) = Ω(µ) = R(µ,A)[e rµ B r + B 0 ]. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 44/50

Now Theorem 8 shows that the system is approximately controllable iff, for µ ρ(a) ρ and x R n, the fact that (µ) 1 Λ(µ)u,x = 0, for u U, (15) implies that x = 0. It can be found that the equality in (15) is equivalent to that (µi A e rµ L r ) 1 (e rµ B r + B 0 )u,x = 0. When B r = 0, this condition is simpler than the one for systems with state delays obtained by Manitius et al in 70s. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 45/50

A rank condition Assume L r = 0. The left-hand side of the equality in (15), assuming µ 0 for the moment, is u (e rµ B r + B 0 ) (µi A) x = u (e rµ B r + B 0 ) Σ n 1 k=0 f k( 1 µ )(Ak ) x = Σ n 1 k=0 u (f k ( 1 µ )e rµ B r + f k ( 1 µ )B 0) (A k ) x where (µi A) 1 is replaced with (µi A) 1 = 1 µ (I 1 µ A) 1 = Σ n 1 k=0 f k( 1 µ )Ak with f k ( 1 µ ), k = 0, 1,n 1, as polynomials in 1 µ. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 46/50

This can be written as uf 0 ( 1 µ )e rµ uf 1 ( 1 µ )e rµ. uf n 1 ( 1 µ )e rµ uf 0 ( 1 µ ) uf 1 ( 1 µ ). uf n 1 ( 1 µ ) B r B r A. Br (A ) n 1 B0 B0 A. B 0 (A ) n 1 2n m n x = 0. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 47/50

Hence, the system is approximately controllable if and only if Rank ( B r AB r A n 1 B r B 0 AB 0 A n 1 B 0 ) = n. The condition µ ρ(a) ρ and µ 0 are now irrelevant. This recovers the results for systems with input delays by Sebakhy (1971). If B r = 0, then Rank ( B 0 AB 0 A n 1 B 0 ) = n. We have not lost ourselves! S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 48/50

A rank condition when L r 0 can be derived similarly. Instead of using A k in the series expansion, (A + e rµ L r ) k, k = 0, 1,n 1, should be used. However, it is difficult to express the condition for the general case in a compact form and hence omitted. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 49/50

Summary The approximate controllability of neutral systems is discussed via converting the problem into that for a perturbed control problem. After solving the approximate controllability for the perturbed control problem, the conditions for the approximate controllability of neutral systems are obtained. Some special cases are discussed. S. HADD AND Q.-C. ZHONG: APPROXIMATE CONTROLLABILITY OF NEUTRAL SYSTEMS WITH STATE AND INPUT DELAYS IN BANACH SPACES p. 50/50