Stabilization of Heat Equation

Transcription

1 Stabilization of Heat Equation Mythily Ramaswamy TIFR Centre for Applicable Mathematics, Bangalore, India CIMPA Pre-School, I.I.T Bombay 22 June - July 4, 215 Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

2 contents 1 Introduction Stabilization- A model 2 Heat Equation - Interior Control Weak Solutions 3 Projected Systems Spectral Analysis Equivalent Projected Systems 4 Stabilization of unstable projected system finite dimensional system Unique Continuation 5 Stabilization of full system 6 Heat equation - Boundary Control Checking of Hautus condition Control of minimal norm Closed loop system Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

3 Introduction One dimensional heat equation Stabilization- A model The heat equation in (, L) (, ) for L > y t y xx = uχ O in (, L) (, ), y(, t) = y(l, t) = in (, ), y(x, ) = y (x) in (, L), Questions : For a given y L 2 (, L), is there a control u with support in a subset O of (, L) such that the solution y L 2 (, ; H 1 (, L)) C([, ); L2 (, L)) decays exponentially in time : y(t) Ke µt y, t? At what rate µ? Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

4 Introduction Stabilization- A model Recall the eigenfunctions φ k and eigenvalues λ k of the homogeneous problem ( ) 2 kπx φ k (x) = L sin, k N, x (, L). L λ k = k2 π 2 L 2. The family (φ k ) k N is a Hilbert basis of L 2 (, L). If A = d2 dx 2 with domain D(A) = H 2 H 1, then φ k D(A), Aφ k = λ k φ k. For the homogeneous problem, the solution is y(x, t) = k=1 y k e k2 π 2 t L 2 φ k (x), x (, L), t (, T ), The function y L 2 (, T ; H 1 (, L)) C([, T ]; L2 (, L)). Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

5 Heat Equation - Interior Control Weak solution of heat equation Weak Solutions Definition (Weak solution of heat equation) Let y L 2 (, L) and f L 2 (, T ; L 2 (, L)). Then y belonging in W (, T ; H 1 (, L), H 1 (, L)) = L 2 (, T ; H 1 (, L) H 1 (, T ; H 1 (, L)), is called a weak solution of the heat equation y t y xx = f in (, L) (, T ), y(, t) = y(l, t) = in (, T ), y(x, ) = y (x) in (, L), if y(, ) = y ( ) and for all φ H 1 (, L), y satisfies d dt L y(t)φ = L y x (t)φ x + L f(t)φ. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

6 Heat Equation - Interior Control Weak Solutions Theorem (Existence and uniqueness of the weak solution) Let y L 2 (, L) and f L 2 (, T ; L 2 (, L)). The heat equation with forcing term f and initial condition y, admits a unique weak solution y and it satisfies y L (,T ;L 2 (,L))+ y L 2 (,T ;H 1 (,L)) C( y L 2 (,L)+ f L 2 (,T ;L 2 (,L))), for some constant C >. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

7 Projected Systems Spectral Analysis We can always choose µ where µ is strictly between two eigenvalues. Let us assume λ 1 < < λ N < µ < λ N+1 <. Define E + = N k=1 E(λ k), E = k=n+1 E(λ k), where E(λ k ) = Ker(λ k I + A), the eigenspace associated to λ k, for all k N. We see that L 2 (, L) = E + E. Let P N be the orthogonal projection of L 2 (, L) onto E +, defined by N P N f = (f, φ k ) L 2 (,L)φ k, k=1 f L 2 (, L). Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

8 Projected Systems Equivalent Projected Systems Theorem Let y be the solution of the heat equation with control u. Then P N y satisfies t P N y xx P N y = P N (uχ O ) in (, L) (, ), P N y(, t) = P N y(l, t) = in (, ), P N y(, ) = P N y ( ) in (, L), and (I P N )y satisfies t (I P N )y xx (I P N )y = (I P N )(uχ O ) in (, L) (, ), (I P N )y(, t) = (I P N )y(l, t) = in (, ), (I P N )y(, ) = (I P N )y ( ) in (, L). Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

9 Projected Systems Equivalent Projected Systems Proof Let y be a solution of the heat equation. We show that P N y satisfies the weak formulation d dt L P N y(t)φ = L P N y x (t)φ x + L P N (uχ O )φ, for all φ E + and so φ = N k=1 (φ, φ k) L 2 (,L)φ k. It follows from the fact that L 2 (, L) is the orthogonal sum Of E + and E, and definition of P N. In fact, we show d dt L y x (t)φ x = L L y(t)φ = d dt P N y x (t)φ x and L L all φ = N k=1 (φ, φ k) L 2 (,L)φ k E +. Similarly, we can show for (I P N )y. P N y(t)φ, uχ O φ = L P N (uχ O )φ, for Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

10 Stabilization of unstable projected system finite dimensional system The system for P N y is in E + for all t and E + is the finite dimensional space. We study the controllability of the finite dimensional system for P N y. Set finite dimensional control for some time T >. Note that u = N v k φ k L 2 (, T ; E + ), k=1 P N (uχ O ) = P N ( N k=1 v kφ k χ O ) N N = v j( φ j φ k )φ k. j=1 k=1 O Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

11 Stabilization of unstable projected system finite dimensional system We introduce B L(R N, E ) such as Bv = N N v j( φ j φ k )φ k, k=1 O j=1 for all v = (v 1,, v N ) T R N. For this u, we write t P N y xx P N y = Bv in (, L) (, ), P N y(, t) = P N y(l, t) = in (, ), P N y(, ) = P N y ( ) in (, L), Setting P N y = y N, A + = P N L(E + ) and P N y = y,n, the above system can be written in E + y N = A + y N + Bv, y N () = y,n E +. Now we want to write an equivalent system in R N. Let y N = N k=1 y kφ k E + and z = (y 1,, y N ) T R N. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

12 Stabilization of unstable projected system finite dimensional system We write where z = z = Λz + Cv, z() = z, Λ = diag(λ 1,, λ N ) R N R N, ( C = φ j φ k )1 j N,1 k N RN R N, O ( (y,n, φ 1 ) L 2 (O),, (y,n, φ N ) L 2 (O)) T R N. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

13 Stabilization of unstable projected system Unique Continuation Theorem The matrix C is invertible if and only if the family {φ k O } 1 k N is linearly independent in L 2 (O). Proof Since the matrix C is the Gram matrix in L 2 (O) of the family {φ k O } k N, the result follows. Theorem The family {φ k O } 1 k N is linearly independent in L 2 (O). Aim To show if N α k φ k O = in L 2 (O), then α k =, k=1 and hence {φ k O } 1 k N is linearly independent in L 2 (O). 1 k N Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

14 Stabilization of unstable projected system Unique Continuation Proof Let N α k φ k O = in L 2 (O). k=1 Denote f = N α k φ k and we have f O =. k=1 Suppose f is in the form f = α k φ k for some k {1, 2,, N}. 2 Since φ k (x) = L sin ( ) kπx L for all x (, L), using this expression we conclude that f = on (, L), if it is given that f O =. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

15 Stabilization of unstable projected system Unique Continuation Now suppose, f = N α k φ k. k=1 Then we reduce it to the first case as follows. We have [λ 1 I + xx ]f = N (λ 1 λ k )α k φ k, k=2 using[λ 1 I + xx ]φ k = (λ 1 λ k )φ k. Thus we eliminate φ 1 from the right hand side. Note that (λ j λ k ), for j k. Similarly, to eliminate φ 1 and φ 2, [λ 2 I + xx ][λ 1 I + xx ]f = N (λ 2 λ k )(λ 1 λ k )α k φ k. k=3 Repeating the above method finitely many times, we can reduce it in the first case and derive the result. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

16 Stabilization of unstable projected system Unique Continuation Theorem The system z = Λz + Cv, z() = z, is exactly controllable at time T, for all T >. Proof By the above two theorems, we have C is invertible. By checking Kalman rank condition, we show the system is controllable. From the above results, it follows Theorem For all y,n E +, there exists a v L 2 (, T ; R N ) such that y N, the solution of the finite dimensional system satisfies y N (T ) = and control v obeys v(t) R N C y,n L 2 (,L), for some constant C >. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

17 Stabilization of full system Theorem For any µ >, there exists a constant K > such that for all initial condition y L 2 (, L), there exists a control u L 2 (, ; L 2 (, L)) such that y y,u, the solution of the heat equation satisfies Proof Set control y y,u(t) L 2 (,L) Ke µt y L 2 (,L), t. u(t) = { N k=1 v k(t)φ k, t T,, t > T, where v k1 k N is obtained from the previous theorem such that P N y(t ) = y N (T ) =. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

18 Stabilization of full system We have v(t) R N C y,n L 2 (,L), where v = (v 1,, v N ) T. We also have P N y(t ) L 2 (,L) C P N y L 2 (,L), for all t T and P N y(t) = for all t T. It yields P N y(t) L 2 (,L) Ce λ N+1t e λ N+1T P N y L 2 (,L), t. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

19 Stabilization of full system Now to estimate the stable part for (A, D(A )) where D(A ) = E D(A) and A y = (I P N )y x x for all y D(A ), we write (I P N )y(t) = e A t (I P N )y + t ( N ) e (t s)a (I P N ) v k (s)φ k O ds. k=1 Using e A t (I P N )y L 2 (,L) e λ N+1t P N y L 2 (,L) and the above estimates, we can show (I P N )y(t) L 2 (,L) Ke µt y L 2 (,L), t. Combining the estimates for P N y and (I P N )y, the theorem follows. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

20 Heat equation - Boundary Control The heat equation in (, 1) (, ) with a boundary control y t y xx = in (, 1) (, ), y(, t) =, y(1, t) = u(t) in (, ), y(x, ) = y (x) in (, 1), where for all t (, ), u(t) R, one dimensional control. We want to write the above equation in a Hilbert space Y as y = Ay + Bu, y() = y Y. B is the control operator and U is a Hilbert space for controls. Consider Y = L 2 (, 1) and U = R, one dimensional space. Recall A = d2 with domain D(A) = H 2 (, 1) H 1 dx 2 (, 1) and D(A) = D(A ) with A = A. φ k is an eigenfunction of A for eigenvalue λ k, where φ k (x) = 2sin (kπx), k N, x (, 1). λ k = k 2 π 2. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

21 Heat equation - Boundary Control Expression of B Aim To determine B and B by the weak formulation. y is a weak solution of the heat equation with nonhomogeneous boundary condition if and only if for all ψ D(A ) = H 2 (, 1) H 1 (, 1), d dt 1 y(x, t)ψ(x)dx = 1 y(x, t)ψ xx (x)dx u(t)ψ x (1). y is a weak solution of the evolution equation if and only if for all ψ D(A ) = H 2 (, 1) H 1 (, 1), d dt 1 y(x, t)ψ(x)dx = 1 y(x, t)aψ(x)dx + (Bu(t), ψ) L 2 (,1). Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

22 Heat equation - Boundary Control Comparing above two identities, we obtain (Bu(t), ψ) L 2 (,1) = u(t)ψ x (1), t (, ) Thus for all t (, ), (u(t), B ψ) R = u(t)ψ x (1) and hence Hautus condition for stabilization: B ψ = ψ x (1). With a prescribed decay ω >, (A + ωi, B) is stabilizable if and only if for all unstable eigenvalues λ k + ω >, implies φ =. A φ = λ k φ, B φ = Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

23 Heat equation - Boundary Control Let choose ω = 1. The only unstable eigenvalue of A + ωi is λ 1 + ω = π >. Unstable space spanned by φ 1, eigenfunction of (A + ωi) for eigenvalue (λ 1 + ω) and E + = Rφ 1, E = k=2 Rφ k, where E is the stable eigenspace. Let us define the projectors associated with this decomposition P 1 f = (f, φ 1 ) L 2 (,1)φ 1, (I P 1 )f = (f, φ k ) L 2 (,1)φ k. Thus k=2 P 1 Bu = (Bu, φ 1 ) L 2 (,1)φ 1 = (u, B φ 1 ) R φ 1 = uφ 1(1)φ 1, (I P 1 )Bu = (Bu, φ k ) L 2 (,1)φ k = u φ k (1)φ k. k=2 Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28 k=2

24 Heat equation - Boundary Control Checking of Hautus condition Using the expression of φ k, we have φ k (1) = 2πk( 1) k. The series k=2 φ k (1)φ k converges in (D(A )), but not in L 2 (, 1). Checking of Hautus condition Recall that only λ 1 + ω >. Let A φ = λ 1 φ and B φ =. Using A = A, φ = cφ 1, for all c R. Since B φ =, we get cφ (1) = 2cπ =. Hence c = and so φ =. Hautus condition is satisfied and (A + ωi, B) is stabilizable. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

25 Heat equation - Boundary Control Control of minimal norm Control of minimal norm To determine control of minimal norm, we project the equation y = (A + ω)y + Bu onto E +. Let P 1 y = y 1 φ 1. The equation for y 1 is y 1 = (1 π2 )y 1 u(t)φ (1) = (1 π 2 )y 1 u(t) 2π, y 1 () = (y, φ 1 ) L 2 (,1). The Bernoulli equation for the system is p >, 2(1 π 2 )p ( π 2) 2 p 2 =. Thus p = 2(1 π2 ) 2π 2. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

26 Heat equation - Boundary Control Control of minimal norm The control of minimal norm obeys the feedback law u(t) = ( π 2) 2(1 π2 ) 2π 2 y 1 (t). y 1 ( ) satisfies the closed loop system y 1 = (1 π 2 )y 1, y 1 () = (y, φ 1 ) L 2 (,1). Hence y 1 (t) = (y, φ 1 ) L 2 (,1)e (1 π2 )t, for all t and so this system is stable. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

27 Heat equation - Boundary Control Closed loop system The closed loop system for ŷ = e 1t y is ŷ t ŷ xx = in (, 1) (, ), ŷ(, t) =, ŷ(1, t) = (π 2) 2(1 π2 ) 2π 2 (ŷ(t), φ 1 ) L 2 (,1) in (, ), ŷ(x, ) = y (x) in (, 1). Since the above system is stabilizable, the closed loop system for y is y t y xx = in (, 1) (, ), y(, t) =, y(1, t) = (π 2) 2(1 π2 ) 2π 2 (y(t), φ 1 ) L 2 (,1) in (, ), y(x, ) = y (x) in (, 1), and y satisfies for some constant C >. y(t) L 2 (,1) Ce 1t y L 2 (,1), Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28

28 Heat equation - Boundary Control Closed loop system A. Bensoussan, G. Da Prato, M. Delfour, S. K. Mitter, Representation and control of infinite dimensional systems. Second edition. Systems and Control: Foundations & Applications. Birkhäuser Boston, Inc., Boston, MA, 27. Lasiecka and Triggiani, Differential and Algebraic Riccati Equations with Applications to Boundary/Point control Problems, Springer-Verlag, 1991 J-L Lions, Optimal Control of systems governed by Partial Differential Equations, Springer, Jean-Pierre Raymond, Optimal Control of PDEs, FICUS Course Notes, 21. Jean-Pierre Raymond, Optimal control and stabilisation of flow related models, FICUS Course Notes, 21. Mythily Ramaswamy Stabilization of Heat Equation 4th July, / 28