Global exact boundary controllability for 1‐D quasilinear wave equations

Based on the local exact boundary controllability for 1‐D quasilinear wave equations, the global exact boundary controllability for 1‐D quasilinear wave equations in a neighborbood of any connected set of constant equilibria is obtained by an extension method. Similar results are also given for a kind of general 1‐D quasilinear hyperbolic equations. Copyright © 2010 John Wiley & Sons, Ltd.     

On approximation theorems for controllability of non-linear parabolic problems

^** Email: anil{at}math.iitb.ac.in^*** Email: mcj{at}math.iitb.ac.in^**** Email: akp{at}math.iitb.ac.in In this paper, we consider the following control system governedby the non-linear parabolic differential equation of the form: [graphic: see PDF] where A is a linear operator with dense domain and f(t, y)is a non-linear function. We have proved that under Lipschitzcontinuity assumption on the non-linear function f(t, y), theset of admissible controls is non-empty. The optimal pair (u^*,y^*) is then obtained as the limit of the optimal pair sequence{(u_n^*, y_n^*)}, where u_n^* is a minimizer of the unconstrainedproblem involving a penalty function arising from the controllabilityconstraint and y_n^* is the solution of the parabolic non-linearsystem defined above. Subsequently, we give approximation theoremswhich guarantee the convergence of the numerical schemes tooptimal pair sequence. We also present numerical experimentwhich shows the applicability of our result.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

A global Carleman inequality and exact controllability of parabolic equations with mixed boundary conditions

This paper establishes a global Carleman inequality of parabolic equations with mixed boundary conditions and an estimate of the solution. Further, we prove exact controllability of the equation by controls acting on an arbitrarily given subdomain or subboundary.     

Approximate controllability of a parabolic integrodifferential equation

In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Regional controllability of semilinear degenerate parabolic equations in bounded domains

In this paper we study controllability properties of semilinear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of ‘regional null controllability’, showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate.     

Controllability of a system of parabolic equations with non-diagonal diffusion matrix

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following systemof parabolic equations with Dirichlet boundary condition: {z_t = D z + b₁(x)u₁ + ··· + b_m(x)u_m, t 0, z ⁿ, z = 0, on where is a sufficiently smooth bounded domain in ^N, b_i L²(;ⁿ), the control functions u_i L²(0, t₁; ); i = 1, 2, ..., mand D is an n x n non-diagonal matrix whose eigenvalues aresemi-simple with positive real part. This algebraic conditionis checkable since it is given in terms of the n_j x m matricesDP_j and P_jB, i.e. Rank [P_jBDP_jBD²P_jB··· D^n_j–1 P_jB]= n_j, where P_jBu = P_jb₁u₁ + ··· + P_jb_mu_m. Finally,this result can be applied to those systems of partial differentialequations that can be rewritten as a diffusion system (see deOliveira, 1998).     

Simultaneous controllability of damped wave equations

We study the exact controllability of q uncoupled damped string equations by means of the same control function. This property is called simultaneous controllability. An observability inequality is proved, which implies the simultaneous controllability of the system. Our results generalize the previous results on the linear wave without the dampings. Copyright © 2016 John Wiley & Sons, Ltd.     

Feedback exact null controllability for unbounded control problems in Hilbert space

Terminal constraint optimal control problems with unbounded control operators are considered. It is shown that the optimal solutions can be represented in a feedback form via a solution of an appropriate Riccati equation. In particular, it is proved that, for systems described by partial differential equations with infinite speed of propagation, boundary exact null controllability can be realized in feedback form.This work was partially supported by the National Science Foundation, Grant No. DMS-89-02811, and by the Air Force Office of Scientific Research, Grant No. AFOSR-89-0511 DEF.     

Asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations. First, for 1D quasi-linear hyperbolic systems with zero eigenvalues, we establish the existence and uniqueness of semiglobal classical solution to the one-sided mixed initial-boundary value problem on a semibounded initial axis and discuss the asymptotic behavior of the corresponding solutions under different hypotheses on the initial data. Based on these results, we obtain the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations on a semibounded time interval.     

Null controllability for a semilinear parabolic equation with gradient quadratic growth

This article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term. The technique in this paper is a combination of Cole–Hopf transformation and some methods from [A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control Optim. 41 (2003) 1886–1900].     

Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.     

Exact controllability of the heat equation with bilinear control

This paper deals with exact controllability of bilinear heat equation. Namely, given the initial state, we would like to provide a class of target states that can be achieved through the heat equation at a finite time by applying multiplicative controls. For this end, an explicit control strategy is constructed. Simulations are provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Eigen series solutions to terminal-state tracking optimal control problems and exact controllability problems constrained by linear parabolic PDEs

Terminal-state tracking optimal control problems for linear parabolic equations are studied in this paper. The control objectives are to track a desired terminal state and the control is of the distributed type. Explicit solution formulae for the optimal control problems are derived in the form of eigen series. Pointwise-in-time L² norm estimates for the optimal solutions are obtained and approximate controllability results are established. Exact controllability is shown when the target state vanishes on the boundary of the spatial domain. One-dimensional computational results are presented which illustrate the terminal-state tracking properties for the solutions expressed by the series formulae.     

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.     

Some results on controllability of a nonlinear degenerate parabolic system by bilinear control

In this paper, we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control. Based on the shrinking property of the solutions, we prove that the system is not globally approximately controllable. Furthermore, we give an approximate null controllability result. We also prove that the system is not globally exactly null controllable by a comparison principle.     

Implicit function theorem and its application to a Ulam’s problem for exact differential equations

We prove a global version of the implicit function theorem under a special condition and apply this result to the proof of a modified Hyers-Ulam-Rassias stability of exact differential equations of the form, g（x, y） ＋ h（x, y）y＇ =0.     

On exact boundary controllability for linearly coupled wave equations

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane.     

Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions

This is a second paper in a two part series. In the prequel, [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.06.101], we showed that a system of Maxwell's equations for a homogeneous medium in a cube with nonnegative conductivity possesses the property that any finite combination of eigenfunctions is controllable (spectral controllability) by means of boundary surface currents applied over only one face of the cube. In the present paper it is established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111-124], that spectral controllability is the strongest result possible for this geometry, since the exact controllability fails regardless of the size of the conductivity term. However, we do establish controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at an appropriate exponential rate. This does not contradict the lack of exact controllability since in any Sobolev space there are initial conditions which violate these restrictions.