##
**Exact controllability and stabilization. The multiplier method.**
*(English)*
Zbl 0937.93003

Research in Applied Mathematics. 36. Chichester: Wiley. Paris: Masson. viii, 156 p. (1994).

The monograph under review contains an excellent summary of the state of the art in exact controllability and stabilizability around the end of the year 1993. It originated from notes of lectures given by the author in Hungary, France, and the U.S.A. It is an excellent, easy introduction to some modern ideas, for example to Hilbert Uniqueness Method (HUM), but at the same time it contains material, that was never published before.

Chapter one starts with a discussion of the small transversal vibrations of an elastic string, a subject that most readers are very familiar with. The mathematical model is represented by the classical system \[ (u_{tt}-u_{xx})(x,t)=0,\quad \{x,t\}\in I\times (0,T), \] with \(u(a,t) = v_a(t)\), \(u(b,t)=v_b(t)\), \(t\in(0,T)\), and \(u(x,0) = u^0(x)\), \(u_t(x,0) = u^1(x)\), \(x\in I\).

This system is called exactly controllable if, given any initial state \((u^0(x)\), \(u^1(x))\), there exist admissible control functions \(v_a(t)\), \(v_b(t)\) such that at the “final” time \(T\) we have: \(u(x,T) = u_t(x,T)=0\). Assuming that the equation does represent the vibration of a physical behavior of a string, we must have the solutions at least continuous on \(I\times (0,T)\). Otherwise distributional solutions may be permitted. Thus \(u\in C([0,T];H^1(I))\cap C^1([0,T];L^2(I))\).

Writing D’Alembert solutions as waves propagating to the right and left with unit velocity one can come to the easy conclusion that the system is exactly controllable if \(T>b-a\), and only approximately controllable if \(T<b-a\). If \(T=b-a\) then the optimal control, which is exact control, is unique. Moreover it is easy to find the feedback law: \(v_a=F_a(u)\), \(v_b=F_b(u)\).

Again the travelling wave Dirichlet solution on the extended interval \((-\infty,+\infty)\) gives us the relations: \((u_x- u_t)(a,t) =(u_x+u_t)(b,t)\) \(\forall t\in \mathbb{R}^+\), with the same boundary conditions on \(u\). It follows that it suffices to solve the travelling wave problem in terms of variables \(\xi=x+t\) and \(\eta=x-t\) in neighborhoods of \(a\) and \(b\), respectively. The author produces the “up”, “down”, “left”, “right” formulas, leaving the final details of this computation to the reader. This discussion of the vibrating string is regarded as an introduction to the fundamental ideas of exact controllability.

The next chapter on linear evolutionary problems begins with a review of reasonably well-known results concerning compact imbedding of a Hilbert space \(V\) in a Hilbert space \(H\) such that \(V\subset H= H'\subset V'\), where primes denote dual spaces. Let \(i\) denote the imbedding of \(V\) in \(V'\). The duality map \(A: V\to V'\) is an isometric isomorphism, by the Riesz representation theorem. The author applies the von Neumann spectral theorem to the operator \(T = A^{-1}\circ i\) to produce a sequence of subspaces \(Z_i\) of \(V\) generated by the eigenspectrum \(\lambda_1,\lambda_2,\dots,\lambda_i\) of \(A\). The union of these spaces is a space \(Z\). Taking an arbitrary vector \(\mathbf v\) and expanding it in orthogonal series converging in \(H\), he assigns the norm on \(Z:\|{\mathbf V}\|^2_{v'}=(\sum \lambda_k^{2\alpha}\|v_k\|^2_H)^{1/2}\). Completion of \(Z\) with respect to this norm is denoted by \(D_\alpha\). Since for \(\alpha>\beta\) the \(\alpha\) norm is stronger than the \(\beta\) norm, the author assumes that a space \(D_\alpha\) can be densely and continuously imbedded in \(D_\beta\), and that such an imbedding is compact. The following properties are listed: For any \(\alpha,\beta\in\mathbb{R}\), \(A^\alpha A^\beta=A^{\alpha+\beta}\), and \(\|v\|_{1/2}=\|v\|_V\), \(\|v\|_0=\|v\|_H\), \(\|v\|_{-1/2}=\|v\|_{V'}\) for any \(v\in Z\). All this has been done before. For example see J. L. Lions and E. Magenes [Problèmes aux limites non-homogènes et applications. Vols. I-III, Dunod, Paris, Vols. 1, 2 (1968; Zbl 0165.10801) and Vol. 3 (1970; Zbl 0197.06701)]. However, the author makes skillful use of these properties, offering fairly easy proofs of existence and/or uniqueness of solutions. Heuristic arguments offering such expansions may be found in engineering papers. The arguments of the author are rigorous and offer a mathematician working in this area an immediate “good” choice of spaces resulting in “clean” energy arguments.

The trigonometric series and the energy expression are written for the homogeneous evolutionary equation: \(u'' + Au =0\).

Next the author considers the wave equation \[ \begin{aligned} & u''+\Delta u+qu=0\quad\text{in}\quad \Omega\times \mathbb{R}: \\ & u=0\quad \text{on }\Gamma\times \mathbb{R},\\ & \partial_\nu u+au=0\quad \text{on }\Gamma_1\times \mathbb{R},\\ & u(0)=u^0,\quad u'(0)=u^1.\end{aligned}\tag{1} \] The appropriate spaces are \(H=L^2\), \(V\), such that \(\|v\|^2_V=\int_\Omega|\nabla v|^2+q|v|^2dx+\int_\Gamma a|v|^2d\Gamma\). Here the imbedding of \(V\) in \(H\) is dense and compact. The corresponding energy is easily derived. Similar computations are completed for Petrovskij systems: \(u''+\Delta^2u=0\), with boundary conditions \(u =\partial_\nu u = 0\), or \(u = 0\) and \(\Delta u = v\) (or 0) on \(\Gamma\times \mathbb{R}\).

The author introduces “the multiplier technique”, which is a variant of the basic ideas of distribution theory and of weak solutions. For example, every term of the wave equation written above is multiplied by \(2h\nabla u\) and integrated by parts, where \(h\) is an arbitrary \(C^1\) vector field.

Several interesting inequalities for the first Petrovskij problem are proved in terms of the first eigenvalue of the related problem \(\Delta^2v =-\mu v\). The idea of Hilbert Uniqueness Method applied to control theory is the use of Hilbert’s uniqueness theorem applied to systems of partial differential equations. Applying this theorem to control theory originated with J. L. Lions and J. Lagnese. The basic theorem from control theory which is exploited here is sufficiency of the observability of the inhomogeneous system to assert exact controllability for the inhomogeneous system. The control applied is of special form. For example in the homogeneous system (1) the control is sought of the form \(v=\partial_\nu u\). The system (1) is compared with the system \[ u'' +\Delta u+qu=0\text{ in }\Omega\times \mathbb{R},\quad y=\partial_\nu u\text{ on }\Gamma_+\times [0,T], \] \(y\) vanishes on the remaining part of the boundary, and \(y(0) = y(T) =0\). This system has a unique solution on \(L^2(\Omega)\times H^{-1}(\Omega)\).

Several uniqueness and controllability results which follow from the HUM are offered in the next chapters. Most are recent, and reflect the recent works of Lions, Zuazua, and the author. Other topics discussed include linear stabilization, including generalization of Russell’s principle that for reversible linear systems stabilizability implies exact controllability. Applications include the wave equation and Maxwell’s equations. Final topics include stabilization of Kirchhoff’s plates, and of the KdV equation.

This book is full of interesting material, such as energy estimates, that appear to be sharper than those previously given by Haraux, whose technique is roughly followed as new inequalities of Lyapunov type are derived.

Chapter one starts with a discussion of the small transversal vibrations of an elastic string, a subject that most readers are very familiar with. The mathematical model is represented by the classical system \[ (u_{tt}-u_{xx})(x,t)=0,\quad \{x,t\}\in I\times (0,T), \] with \(u(a,t) = v_a(t)\), \(u(b,t)=v_b(t)\), \(t\in(0,T)\), and \(u(x,0) = u^0(x)\), \(u_t(x,0) = u^1(x)\), \(x\in I\).

This system is called exactly controllable if, given any initial state \((u^0(x)\), \(u^1(x))\), there exist admissible control functions \(v_a(t)\), \(v_b(t)\) such that at the “final” time \(T\) we have: \(u(x,T) = u_t(x,T)=0\). Assuming that the equation does represent the vibration of a physical behavior of a string, we must have the solutions at least continuous on \(I\times (0,T)\). Otherwise distributional solutions may be permitted. Thus \(u\in C([0,T];H^1(I))\cap C^1([0,T];L^2(I))\).

Writing D’Alembert solutions as waves propagating to the right and left with unit velocity one can come to the easy conclusion that the system is exactly controllable if \(T>b-a\), and only approximately controllable if \(T<b-a\). If \(T=b-a\) then the optimal control, which is exact control, is unique. Moreover it is easy to find the feedback law: \(v_a=F_a(u)\), \(v_b=F_b(u)\).

Again the travelling wave Dirichlet solution on the extended interval \((-\infty,+\infty)\) gives us the relations: \((u_x- u_t)(a,t) =(u_x+u_t)(b,t)\) \(\forall t\in \mathbb{R}^+\), with the same boundary conditions on \(u\). It follows that it suffices to solve the travelling wave problem in terms of variables \(\xi=x+t\) and \(\eta=x-t\) in neighborhoods of \(a\) and \(b\), respectively. The author produces the “up”, “down”, “left”, “right” formulas, leaving the final details of this computation to the reader. This discussion of the vibrating string is regarded as an introduction to the fundamental ideas of exact controllability.

The next chapter on linear evolutionary problems begins with a review of reasonably well-known results concerning compact imbedding of a Hilbert space \(V\) in a Hilbert space \(H\) such that \(V\subset H= H'\subset V'\), where primes denote dual spaces. Let \(i\) denote the imbedding of \(V\) in \(V'\). The duality map \(A: V\to V'\) is an isometric isomorphism, by the Riesz representation theorem. The author applies the von Neumann spectral theorem to the operator \(T = A^{-1}\circ i\) to produce a sequence of subspaces \(Z_i\) of \(V\) generated by the eigenspectrum \(\lambda_1,\lambda_2,\dots,\lambda_i\) of \(A\). The union of these spaces is a space \(Z\). Taking an arbitrary vector \(\mathbf v\) and expanding it in orthogonal series converging in \(H\), he assigns the norm on \(Z:\|{\mathbf V}\|^2_{v'}=(\sum \lambda_k^{2\alpha}\|v_k\|^2_H)^{1/2}\). Completion of \(Z\) with respect to this norm is denoted by \(D_\alpha\). Since for \(\alpha>\beta\) the \(\alpha\) norm is stronger than the \(\beta\) norm, the author assumes that a space \(D_\alpha\) can be densely and continuously imbedded in \(D_\beta\), and that such an imbedding is compact. The following properties are listed: For any \(\alpha,\beta\in\mathbb{R}\), \(A^\alpha A^\beta=A^{\alpha+\beta}\), and \(\|v\|_{1/2}=\|v\|_V\), \(\|v\|_0=\|v\|_H\), \(\|v\|_{-1/2}=\|v\|_{V'}\) for any \(v\in Z\). All this has been done before. For example see J. L. Lions and E. Magenes [Problèmes aux limites non-homogènes et applications. Vols. I-III, Dunod, Paris, Vols. 1, 2 (1968; Zbl 0165.10801) and Vol. 3 (1970; Zbl 0197.06701)]. However, the author makes skillful use of these properties, offering fairly easy proofs of existence and/or uniqueness of solutions. Heuristic arguments offering such expansions may be found in engineering papers. The arguments of the author are rigorous and offer a mathematician working in this area an immediate “good” choice of spaces resulting in “clean” energy arguments.

The trigonometric series and the energy expression are written for the homogeneous evolutionary equation: \(u'' + Au =0\).

Next the author considers the wave equation \[ \begin{aligned} & u''+\Delta u+qu=0\quad\text{in}\quad \Omega\times \mathbb{R}: \\ & u=0\quad \text{on }\Gamma\times \mathbb{R},\\ & \partial_\nu u+au=0\quad \text{on }\Gamma_1\times \mathbb{R},\\ & u(0)=u^0,\quad u'(0)=u^1.\end{aligned}\tag{1} \] The appropriate spaces are \(H=L^2\), \(V\), such that \(\|v\|^2_V=\int_\Omega|\nabla v|^2+q|v|^2dx+\int_\Gamma a|v|^2d\Gamma\). Here the imbedding of \(V\) in \(H\) is dense and compact. The corresponding energy is easily derived. Similar computations are completed for Petrovskij systems: \(u''+\Delta^2u=0\), with boundary conditions \(u =\partial_\nu u = 0\), or \(u = 0\) and \(\Delta u = v\) (or 0) on \(\Gamma\times \mathbb{R}\).

The author introduces “the multiplier technique”, which is a variant of the basic ideas of distribution theory and of weak solutions. For example, every term of the wave equation written above is multiplied by \(2h\nabla u\) and integrated by parts, where \(h\) is an arbitrary \(C^1\) vector field.

Several interesting inequalities for the first Petrovskij problem are proved in terms of the first eigenvalue of the related problem \(\Delta^2v =-\mu v\). The idea of Hilbert Uniqueness Method applied to control theory is the use of Hilbert’s uniqueness theorem applied to systems of partial differential equations. Applying this theorem to control theory originated with J. L. Lions and J. Lagnese. The basic theorem from control theory which is exploited here is sufficiency of the observability of the inhomogeneous system to assert exact controllability for the inhomogeneous system. The control applied is of special form. For example in the homogeneous system (1) the control is sought of the form \(v=\partial_\nu u\). The system (1) is compared with the system \[ u'' +\Delta u+qu=0\text{ in }\Omega\times \mathbb{R},\quad y=\partial_\nu u\text{ on }\Gamma_+\times [0,T], \] \(y\) vanishes on the remaining part of the boundary, and \(y(0) = y(T) =0\). This system has a unique solution on \(L^2(\Omega)\times H^{-1}(\Omega)\).

Several uniqueness and controllability results which follow from the HUM are offered in the next chapters. Most are recent, and reflect the recent works of Lions, Zuazua, and the author. Other topics discussed include linear stabilization, including generalization of Russell’s principle that for reversible linear systems stabilizability implies exact controllability. Applications include the wave equation and Maxwell’s equations. Final topics include stabilization of Kirchhoff’s plates, and of the KdV equation.

This book is full of interesting material, such as energy estimates, that appear to be sharper than those previously given by Haraux, whose technique is roughly followed as new inequalities of Lyapunov type are derived.

Reviewer: Vadim Komkov (Florida)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93B05 | Controllability |

93C20 | Control/observation systems governed by partial differential equations |

93D15 | Stabilization of systems by feedback |