M. Slemrod

Bio: M. Slemrod is an academic researcher from Rensselaer Polytechnic Institute. The author has contributed to research in topics: Semigroup & Infinitesimal generator. The author has an hindex of 3, co-authored 5 publications receiving 493 citations.

Controllability for Distributed Bilinear Systems

Abstract: This paper studies controllability of systems of the form ${{dw} / {dt}} = \mathcal {A}w + p(t)\mathcal {B}w$ where $\mathcal{A}$ is the infinitesimal generator of a $C^0$ semigroup of bounded linear operators $e^{\mathcal{A}t} $ on a Banach space X, $\mathcal{B}:X \to X$ is a $C^1$ map, and $p \in L^1 ([0,T];\mathbb{R})$ is a control. The paper (i) gives conditions for elements of X to be accessible from a given initial state $w_0$ and (ii) shows that controllability to a full neighborhood in X of $w_0$ is impossible for $\dim X = \infty $. Examples of hyperbolic partial differential equations are provided.

Feedback Stabilization of Distributed Semilinear Control Systems

Abstract: This paper considers feedback stabilization for the semilinear control system\(\dot u(t) = Au(t) + \upsilon (t)B(u(t)).\) HereA is the infinitesimal generator of a linearC0 semigroup of contractions on a Hilbert spaceH andB : H → H is a nonlinear operator. A sufficient condition for feedback stabilization is given and applications to hyperbolic boundary value problems are presented.

Scanning control of a vibrating string

Abstract: We construct scanning feedback controls {γ i (t)} for the vibrating string equation $$\begin{gathered} y_{tt} (x,t) = y_{xx} (x,t) + Ry(x,t) + \sum\limits_{i = 1}^N {\phi (x - \gamma _i } (t))y(x,t), \hfill \\ 0< x< 1,y = 0 at x = 0,1. \hfill \\ \end{gathered} $$ so that (y, y t ) → (0,0) ast → ∞ in the weak topology ofH 0 1 (0,1) ×L 2 (0,1). In particular we show that ifφ is an even polynomial of degreeN with nonpositive coefficients that forR <π 2 we can find such stabilizingγ i (t), i=1,⋯,N.

Controllability for a class of nondiagonal hyperbolic distributed bilinear systems

Abstract: This paper studies controllability of the abstract “hyperbolic” control systemu + Au + p(t)Bu = 0 whereA andB are (possibly) unbounded linear operators on an infinite dimensional Hilbert spaceH andp(t) is a real valued scalar control. The paper gives conditions for elements of the underlying state space to be accessible from prescribed initial data (u(0),57-1 Applications to wave equations are provided.

Controllability and stabilizability of distributed bilinear systems: Recent results and open problems

Abstract: This paper describes recent results for controlling and stabilizing control systems of the form u(t) = Au(t) + p(t) B(u(t)) where A is the infinitesimal generator C∞ semigroup on a Banach space X, B' map from X into X, and p(t) is a real valued control. Application to a vibrating beam problem is given for illusstration of the theory.

Trends in large space structure control theory: Fondest hopes, wildest dreams

Abstract: This paper presents a mathematical framework for discussion of large space structure (LSS) control theory. Within this framework, current trends in LSS control theory and related topics in general control science are surveyed.

Training Schrödinger’s cat: quantum optimal control

Abstract: It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems We address key challenges and sketch a roadmap for future developments

On the controllability of quantum‐mechanical systems

Abstract: The systems‐theoretic concept of controllability is elaborated for quantum‐mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space H of the system. The Schrodinger equation for a quantum object influenced by adjustable external fields provides a state‐evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain Dω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite‐dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of Dω with a suitably defined finite‐dimensional submanifold of the unit sphere SH in H. Several soluble examples are presented to illuminate th...

Notions of controllability for bilinear multilevel quantum systems

Abstract: In this note, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability.