We discuss several new results on nonnegative approximate controllability for the one-dimensional Heat equation governed by either multiplicative or nonnegative additive control, acting within a proper subset of the space domain at every moment of time. Our methods allow us to link these two types of controls to some extend. The main results include approximate controllability properties both for the static and mobile control supports.
This paper studies controllability of a discrete-time linear dynamical system using nonnegative and sparse inputs. These constraints on the control input arise naturally in many real-life systems where the external influence on the system is unidirectional, and activating each input node adds to the cost of control. We derive the necessary and sufficient conditions for controllability of the system, without imposing any constraints on the system matrices. Unlike the well-known Kalman rank based controllability criteria, the conditions presented in this paper can be verified in polynomial time,…
Let us consider a nonlinear degenerate reaction-diffusion equation with application to climate science. After proving that the solution remains nonnegative at any time, when the initial state is nonnegative, we prove the approximate controllability between nonnegative states at any time via multiplicative controls, that is, using as control the reaction coefficient.
In this paper we establish several results on approximate controllability of a semilinear wave equation by making use of a single multiplicative control. These results are then applied to discuss the exact controllability properties for the one dimensional version of the system at hand. The proof relies on linear semigroup theory and the results on the additive controllability of the semilinear wave equation. The approaches are constructive and provide explicit steering controls. Moreover, in the context of undamped wave equation, the exact controllability is established for a time which is un…
We consider the semilinear heat equation posed on a smooth bounded domain $Ω$ of $\mathbb{R}^{N}$ with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset $ω$ of $Ω$. The goal of this paper is to prove the uniform large time global null-controllability for semilinearities $f(s) = \pm |s| \log^α(2+|s|)$ where $α\in [3/2,2)$ which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global …
In this paper we are concerned with the approximate controllability of a multidimensional semilinear reaction-diffusion equation governed by a multiplicative control, which is locally distributed in the reaction term. For a given initial state we provide sufficient conditions on the desirable state to be approximately reached within an arbitrarily small time interval. Moreover, in the case of a globally supported control, we prove the approximate controllability within any time-interval given in advance which does not depend on the initial and target states. Our approaches are based on linear …
One proves that the $n$-D stochastic controlled equation $dX+AXdt=σ(X)dW+Bu\,dt$, where $σ\in\mbox{Lip}((\R^n,Ł(\R^d,\R^n))$ and the pair $A\inŁ(\R^n)$, $B\inŁ(\R^m,\R^n)$ satisfies the Kalman rank condition, is exactly controllable in each $y\in\R^n$, $σ(y)=0$ on each finite interval $(0,T)$. An application to approximate controllability to stochastic heat equation is given.
This paper aims at the study of controllability properties and induced controllability metrics on complex networks governed by a class of (discrete time) linear decision processes with mul-tiplicative noise. The dynamics are given by a couple consisting of a Markov trend and a linear decision process for which both the "deterministic" and the noise components rely on trend-dependent matrices. We discuss approximate, approximate null and exact null-controllability. Several examples are given to illustrate the links between these concepts and to compare our results with their continuous-time cou…
We study the global approximate controllability of the reaction-diffusion equation in a parallelpiped $ Ω= (a_1,b_1 ) \times \ldots (a_n,b_n) \subset R^n $, governed by a multiplicative control in a reaction term. It is assumed that the initial state $ u_0 $ admits zeros only on the intersections of $ Ω$ with finitely many hyperplanes, parallel to the sides of $ Ω$, and that $ u_0$ changes its sign after crossing such hyperplanes (we further refer to them as the "hyperplanes of change of sign" or "zero hyperplanes"). This paper can be viewed as a continuation of work presented in \cite{CanKh, …
In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems. First, we will obtain embedding results for weighted Sobolev spaces, that have proved decisive in reaching well-posedness for nonlinear degenerate problems. Then, we show that the above systems can be steered in $L^2$ from any nonzero, nonnegative initial state into any neighborhood of any desirable nonnegative target-state by bilinear piecewise static controls. Moreover, we extend the above result relaxing the sign constraint on the initial date.
We study the global approximate controllability properties of a one dimensional semilinear reaction-diffusion equation governed via the coefficient of the reaction term. It is assumed that both the initial and target states admit no more than finitely many changes of sign. Our goal is to show that any target state $ u^*\in H_0^1 (0,1)$, with as many changes of sign in the same order as the given initial data $ u_0\in H^1_0(0,1)$, can be approximately reached in the $ L^2 (0,1)$-norm at some time $T>0$. Our method employs shifting the points of sign change by making use of a finite sequence of …
We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, when the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved betwee…
In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied …
The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and c…
Nonnegative matrix factorization (NMF) is a data analysis technique used in a great variety of applications such as text mining, image processing, hyperspectral data analysis, computational biology, and clustering. In this paper, we consider two well-known algorithms designed to solve NMF problems, namely the multiplicative updates of Lee and Seung and the hierarchical alternating least squares of Cichocki et al. We propose a simple way to significantly accelerate these schemes, based on a careful analysis of the computational cost needed at each iteration, while preserving their convergence p…
In this paper we study the finite-horizon optimal covariance steering problem for a continuous-time linear stochastic system subject to both additive and multiplicative noise. The noise can be continuous or it may contain jumps. Additive noise does not depend on the state or the control, whereas multiplicative noise has a magnitude proportional to the current state. The cost is assumed to be quadratic in both the state and the control. First, the controllability of the state covariance is established under mild assumptions. Then, the optimal control for steering the covariance is provided. Las…
We study the controllability of a coupled system of linear parabolic equations, with non-negativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an "approximate" nonnegativity constraint, and a another stronger one, with "exact" nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a "staircase" method. Finally, we show that state-constrained controllability admits a positive minimal time…
In this paper we study the exact null-controllability property for a class of controlled PDMP of switch type with switch-dependent, piecewise linear dynamics and multiplicative jumps. First, we show that exact null-controllability induces a con-trollability metric. This metric is linked to a class of backward stochastic Riccati equations. Using arguments similar to the euclidian-valued BSDE in [4], the equation is shown to be equivalent to an iterative family of deterministic Riccati equations that are solvable. Second, we give an example showing that, for switch-dependent coefficients, exact …
We propose an explicit, easily-computable algebraic criterion for approximate null-controllability of a class of general piecewise linear switch systems with multiplicative noise. This gives an answer to the general problem left open in [13]. The proof relies on recent results in [4] allowing to reduce the dual stochastic backward system to a family of ordinary differential equations. Second, we prove by examples that the notion of approximate controllability is strictly stronger than approximate null-controllability. A sufficient criterion for this stronger notion is also provided. The result…
Carleman Estimates and Controllability of Stochastic degenerate parabolic Heat Equations
Cadi Ayyad University（卡迪·阿亚德大学）；Faculty of Sciences Semlalia（塞姆拉利亚理学院）；Laboratory of Mathematics, Modeling and Automatic Systems（数学、建模与自动系统实验室）；Ecole Normale Supérieure, Université Cadi Ayyad（卡迪·阿亚德大学高等师范学院）；Cadi Ayyad University, National School of Applied Sciences（卡迪·阿亚德大学国立应用科学学院）；Laboratory of modelisation of complex systems (LMCS)（复杂系统建模实验室）
Université de Mons（蒙斯大学）；Ecole Centrale de Nantes（南特中央理工学院）
Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as the product of two smaller nonnegative matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $β$-divergences ($β$-NMF) with disjoint equality constraints, and with penalty terms in the objective function. By disjoint, we mean that each variable appears in at most one equality constraint. Our MU satisfy the set of constraints after each update of the variables during the optimization process, while guaranteeing that t…
This paper deals with some reachability issues for piecewise linear switched systems with time-dependent coefficients and multiplicative noise. Namely, it aims at characterizing data that are almost reachable at some fixed time T > 0 (belong to the closure of the reachable set in a suitable L 2-sense). From a mathematical point of view, this provides the missing link between approximate controllability towards 0 and approximate controllability towards given targets. The methods rely on linear-quadratic control and Riccati equations. The main novelty is that we consider an LQ problem with contr…
We study the exact controllability of the evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution…
We consider the nonlinear Schrödinger equation (NLS) on a torus of arbitrary dimension. The equation is studied in presence of an external potential field whose time-dependent amplitude is taken as control. Assuming that the potential satisfies a saturation property, we show that the NLS equation is approximately controllable between any pair of eigenstates in arbitrarily small time. The proof is obtained by developing a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. We give an application of this result to the study of the large time behavior of th…
UCLA（加州大学洛杉矶分校）；Colorado State University（科罗拉多州立大学）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；CSU Channel Islands（加州州立大学海峡群岛分校）；Wofford College（沃福德学院）；NYU（纽约大学）；University of Alabama（阿拉巴马大学）；Boston Medical Center（波士顿医疗中心）；Southern Methodist University（南卫理公会大学）；Occidental College（occidental学院）
We propose several new models for semi-supervised nonnegative matrix factorization (SSNMF) and provide motivation for SSNMF models as maximum likelihood estimators given specific distributions of uncertainty. We present multiplicative updates training methods for each new model, and demonstrate the application of these models to classification, although they are flexible to other supervised learning tasks. We illustrate the promise of these models and training methods on both synthetic and real data, and achieve high classification accuracy on the 20 Newsgroups dataset.
