Approximation of the exact controls for the beam equation Sorin - PowerPoint PPT Presentation

Approximation of the exact controls for the beam equation Sorin Micu University of Craiova (Romania) Graz, June 25, 2015 Joint works with Florin Bugariu, Nicolae Cˆ ındea, Ionel Rovent ¸a and Laurent ¸iu Temereanc˘ a

Controlled hinged beam equation Given any time T > 0 and initial data ( u 0 , u 1 ) ∈ H := H 1 0 (0 , π ) × H − 1 (0 , π ) , the exact controllability in time T of the linear beam equation with hinged (simply-supported) ends,   u ′′ ( t, x ) + u xxxx ( t, x ) = 0 , x ∈ (0 , π ) , t > 0     u ( t, 0) = u ( t, π ) = u xx ( t, 0) = 0 , t > 0 (1)  u xx ( t, π ) = v ( t ) , t > 0     u (0 , x ) = u 0 ( x ) , u ′ (0 , x ) = u 1 ( x ) , x ∈ (0 , π ) consists of finding a scalar function v ∈ L 2 (0 , T ) , called control, such that the corresponding solution ( u, u ′ ) of (1) verifies u ( T, · ) = u ′ ( T, · ) = 0 . (2)

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method)

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method) Multipliers

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method) Multipliers Carleman estimates

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method) Multipliers Carleman estimates Microlocal Analysis

(Many) methods to study the controllability Several approaches are available for the study of a controllability problem: Moment theory Direct methods Transmutation methods Uniform stabilization Optimization methods (Hilbert Uniqueness Method) Multipliers Carleman estimates Microlocal Analysis Fattorini H. O. and Russell D. L., Exact controllability theorems for linear parabolic equations in one space dimension , Arch. Rat. Mech. Anal., 4 (1971), 272-292. J.-L. Lions, Controlabilit´ e exacte, stabilisation et perturbations des syst` emes distribu´ es , Vol. 1, Masson, Paris, 1988.

Optimization method Lemma Let T > 0 and ( u 0 , u 1 ) ∈ H . The function v ∈ L 2 (0 , T ) is a control which drives to zero the solution of (1) in time T if and only if, for any ( ϕ 0 , ϕ 1 ) ∈ H , � T � � � � u 1 ( x ) , ϕ (0 , x ) u 0 ( x ) , ϕ ′ (0 , x ) v ( t ) ϕ x ( t, 1) dt = − − 1 , 1 + 1 , − 1 , 0 where ( ϕ, ϕ ′ ) ∈ H is the solution of the backward equation  ϕ ′′ ( t, x ) + ϕ xxxx ( t, x ) = 0 ( t, x ) ∈ (0 , T ) × (0 , 1)     ϕ ( t, 0) = ϕ ( t, 1) = 0 t ∈ (0 , T )  ϕ xx ( t, 0) = ϕ xx ( t, 1) = 0 t ∈ (0 , T ) (3)   ϕ ( T, x ) = ϕ 0 ( x ) x ∈ (0 , 1)    ϕ ′ ( T, x ) = ϕ 1 ( x ) x ∈ (0 , 1) .

Optimization method For each ( u 0 , u 1 ) ∈ H , define the functional J : H → R , � T � � � � J ( ϕ 0 , ϕ 1 ) = 1 | ϕ x ( t, 1) | 2 dt + u 1 ( x ) , ϕ (0 , x ) u 0 ( x ) , ϕ ′ (0 , x ) − 1 , 1 − 1 , − 1 , 2 0 where ( ϕ, ϕ ′ ) is the solution of (3) with initial data ( ϕ 0 , ϕ 1 ) .

Optimization method For each ( u 0 , u 1 ) ∈ H , define the functional J : H → R , � T � � � � J ( ϕ 0 , ϕ 1 ) = 1 | ϕ x ( t, 1) | 2 dt + u 1 ( x ) , ϕ (0 , x ) u 0 ( x ) , ϕ ′ (0 , x ) − 1 , 1 − 1 , − 1 , 2 0 where ( ϕ, ϕ ′ ) is the solution of (3) with initial data ( ϕ 0 , ϕ 1 ) . ϕ 0 , � ϕ 1 ) ∈ H then � If J has a minimum at ( � v ( t ) = � ϕ x (1 , t ) is a control for (1).

Optimization method For each ( u 0 , u 1 ) ∈ H , define the functional J : H → R , � T � � � � J ( ϕ 0 , ϕ 1 ) = 1 | ϕ x ( t, 1) | 2 dt + u 1 ( x ) , ϕ (0 , x ) u 0 ( x ) , ϕ ′ (0 , x ) − 1 , 1 − 1 , − 1 , 2 0 where ( ϕ, ϕ ′ ) is the solution of (3) with initial data ( ϕ 0 , ϕ 1 ) . ϕ 0 , � ϕ 1 ) ∈ H then � If J has a minimum at ( � v ( t ) = � ϕ x (1 , t ) is a control for (1). J has a minimum if it is coercive and it is coercive if the following observability inequality holds for any ( ϕ 0 , ϕ 1 ) ∈ H : � T � ( ϕ (0) , ϕ ′ (0)) � 2 | ϕ x ( t, π ) | 2 dt. H ≤ C (4) 0

Optimization method For each ( u 0 , u 1 ) ∈ H , define the functional J : H → R , � T � � � � J ( ϕ 0 , ϕ 1 ) = 1 | ϕ x ( t, 1) | 2 dt + u 1 ( x ) , ϕ (0 , x ) u 0 ( x ) , ϕ ′ (0 , x ) − 1 , 1 − 1 , − 1 , 2 0 where ( ϕ, ϕ ′ ) is the solution of (3) with initial data ( ϕ 0 , ϕ 1 ) . ϕ 0 , � ϕ 1 ) ∈ H then � If J has a minimum at ( � v ( t ) = � ϕ x (1 , t ) is a control for (1). J has a minimum if it is coercive and it is coercive if the following observability inequality holds for any ( ϕ 0 , ϕ 1 ) ∈ H : � T � ( ϕ (0) , ϕ ′ (0)) � 2 | ϕ x ( t, π ) | 2 dt. H ≤ C (4) 0 Hence, if (4) holds, for any initial data ( u 0 , u 1 ) ∈ H , there exists a control v ∈ L 2 (0 , T ) with the property √ C � ( u 0 , u 1 ) � H . � v � L 2 ≤ (5)

Ingham’s inequality Observability inequality (4) is equivalent to inequality of the form � � � 2 � � � T � � � | α n | 2 ≤ C ( T ) 2 dt, ( α n ) n ∈ Z ∗ ∈ ℓ 2 . α n e ν n t � � (6) � � − T n ∈ Z ∗ n ∈ Z ∗ 2 Ingham’s inequality For any T > 2 π γ ∞ , γ ∞ = lim inf n →∞ | ν n +1 − ν n | , inequality (6) holds. A. E. Ingham, Some trigonometric inequalities with applications to the theory of series, Math. Zeits., 41 (1936), 367-379. J. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Comm. Pure Appl. Math., 32 (1979), 555-587. J. P. Kahane: Pseudo-P´ eriodicit´ e et S´ eries de Fourier Lacunaires, Ann. Sci. Ecole Norm. Super. 37, 93-95 (1962).

Observability inequality In our particular case ν n = i sgn ( n ) n 2 , γ ∞ = lim inf n →∞ | ν n +1 − ν n | = ∞ . Ingham’s inequality implies that the observability inequality (4) is verified for any T > 0 . Consequently, given any T > 0 , there exists a control v ∈ L 2 (0 , T ) for each ( u 0 , u 1 ) ∈ H . The control function v is not unique.

Moment problem for the beam equation The null-controllability of the beam equation is equivalent to solve a moment problem. Lemma Let T > 0 and �� ∞ � n sin( nx ) , � ∞ ( u 0 , u 1 ) = n =1 a 0 n =1 a 1 n sin( nx ) ∈ H . The function v ∈ L 2 (0 , T ) is a control which drives to zero the solution of (1) in time T if and only if � � � T e tν n dt = ( − 1) n e − T 2 ν n � � t + T 2 ν n a 0 n − a 1 ( n ∈ Z ∗ ) , v √ n 2 2 nπ − T 2 (7) where ν n = i sgn ( n ) n 2 are the eigenvalues of the unbounded skew-adjoint differential operator corresponding to (1). A solution v of the moment problem may be constructed by means of a biorthogonal family to the sequence ( e ν n t ) n ∈ Z ∗ .

Moment problem for the beam equation Definition A family of functions ( φ m ) m ∈ Z ∗ ⊂ L 2 � � − T 2 , T with the property 2 � T 2 φ m ( t ) e ν n t dt = δ mn ∀ m, n ∈ Z ∗ , (8) − T 2 is called a biorthogonal sequence to ( e ν n t ) n ∈ Z ∗ in L 2 � � − T 2 , T . 2

Moment problem for the beam equation Definition A family of functions ( φ m ) m ∈ Z ∗ ⊂ L 2 � � − T 2 , T with the property 2 � T 2 φ m ( t ) e ν n t dt = δ mn ∀ m, n ∈ Z ∗ , (8) − T 2 is called a biorthogonal sequence to ( e ν n t ) n ∈ Z ∗ in L 2 � � − T 2 , T . 2 Once we have a biorthogonal sequence to ( e ν n t ) n ∈ Z ∗ , a “formal” solution of the moment problem is given by � � � ( − 1) n e − T 2 ν n � � t − T ν n a 0 n − a 1 √ v ( t ) = φ n . (9) n 2 2 nπ n ∈ Z ∗