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We study the turnpike phenomenon for optimal control problems with mean-field dynamics that are obtained as the limit $N\rightarrow \infty$ of systems governed by a large number $N$ of ordinary differential equations. We show that the optimal control problems with large time horizons give rise to a turnpike structure of the optimal state and the optimal control. For the proof, we use the fact that the turnpike structure for the problems on the level of ordinary differential equations is preserved under the corresponding mean-field limit. [ABSTRACT FROM AUTHOR]

The chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno's formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno's formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000). [ABSTRACT FROM AUTHOR]

Exponential stabilization to time-dependent trajectories for the incompressible Navier–Stokes equations is achieved with explicit feedback controls. The fluid is contained in a given two-dimensional spatial domain. An appropriate finite number of actuators is constructed and the control force is, at each time instant, a linear combination of such actuators. Each actuator has its vorticity supported in a small subdomain. The velocity field is subject to Lions boundary conditions. Simulations are presented showing the stabilizing performance of the proposed feedback. The results also apply to a class of observer design problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the exact boundary controllability of nodal profile for nonautonomous quasilinear hyperbolic systems on a finite time interval and obtain that there is the exact boundary controllability of nodal profile for any given initial time t 0 ∈ R so long as the controllability time T > 0 is suitably large, but T depends on t 0 in general. As applications, we discuss the exact boundary controllability of nodal profile for the Saint-Venant system with a nonuniform subcritical steady state and the system of traffic flow depending on time. [ABSTRACT FROM AUTHOR]

An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heat-structure interaction partial differential equation (PDE) models, with structure subject to high Kelvin-Voigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant fluid-structure models, with wave- or plate-structures, possibly without Kelvin-Voigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDE-energy estimates dictate how to define the interface/boundary feedback control in each of the three cases.

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the performance output tracking for a Schrödinger PDE-ODE cascaded system with external disturbances enter in all possible channels. The main challenge of the problem is the fact that the disturbances are non-collocated to the controller. By proper trajectory planning approach, this difficulty can be overcome by converting non-collocated configurations into the collocated ones. Then a state observer is designed in terms of the tracking errors. Finally, the feedback control is proposed by applying the backstepping technique. The stability of the closed-loop system and the exponential convergence of the regulation error are proved. Some numerical simulations illustrate that the proposed approach is very effective. [ABSTRACT FROM AUTHOR]

This paper considers the feedback stabilization of infinite-dimensional semi-linear systems with discrete multi-delays. Sufficient conditions for an appropriate sequence of feedback control are given to ensure exponential and weak stabilization of the considered system. Moreover, applications with numerical simulation to wave equations and heat equation are presented. [ABSTRACT FROM AUTHOR]

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary to collect coarse mesh measurements of all the state variables that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence, the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information on the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; i.e. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

In this paper, we consider an acoustic equation with incomplete data, where the bulk modulus coefficient and initial conditions are partially known. Our goal is to get information about the bulk modulus coefficient independently of the initial conditions from boundary observations. To achieve this goal, we apply the sentinel method introduced by J.L. Lions, which is a functional that links the solution to the given problem with a control function and a state observation. We prove that the existence of the sentinel functional is equivalent to a boundary-null controllability problem with constraints on the control. We use the Hilbert uniqueness method to study this controllability problem to establish the control of minimal norm. [ABSTRACT FROM AUTHOR]

In this paper we show the global well-posedness of solutions to a three-dimensional magnetohydrodynamical (MHD) model in porous media. Compared to the classical MHD equations, our system involves a nonlinear damping term in the momentum equations due to the "Brinkman-Forcheimerextended-Darcy" law of ow in porous media.

In this paper we survey the various implementations of a new data assimilation (downscaling) algorithm based on spatial coarse mesh measurements. As a paradigm, we demonstrate the application of this algorithm to the 3D Leray-α subgrid scale turbulence model. Most importantly, we use this paradigm to show that it is not always necessary to collect coarse mesh measurements of all the state variables that are involved in the underlying evolutionary system, in order to recover the corresponding exact reference solution. Specifically, we show that in the case of the 3D Leray-α model of turbulence, the solutions of the algorithm, constructed using only coarse mesh observations of any two components of the three-dimensional velocity field, and without any information on the third component, converge, at an exponential rate in time, to the corresponding exact reference solution of the 3D Leray-α model. This study serves as an addendum to our recent work on abridged continuous data assimilation for the 2D Navier-Stokes equations. Notably, similar results have also been recently established for the 3D viscous Planetary Geostrophic circulation model in which we show that coarse mesh measurements of the temperature alone are sufficient for recovering, through our data assimilation algorithm, the full solution; i.e. the three components of velocity vector field and the temperature. Consequently, this proves the Charney conjecture for the 3D Planetary Geostrophic model; namely, that the history of the large spatial scales of temperature is sufficient for determining all the other quantities (state variables) of the model.

We describe a spectrally filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.

We consider fully discrete numerical schemes for a downscaling dataassimilation algorithm aimed at approximating the velocity field of the 2DNavier-Stokes equations corresponding to given coarse mesh observationalmeasurements. The time discretization is done by considering semi- andfully-implicit Euler schemes, and the spatial discretization is based on aspectral Galerkin method. The two fully discrete algorithms are shown to beunconditionally stable, with respect to the size of the time step, number oftime steps and the number of Galerkin modes. Moreover, explicit, uniform intime error estimates between the fully discrete solution and the referencesolution corresponding to the observational coarse mesh measurements areobtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensionalNavier-Stokes equations, subject to the no-slip Dirichlet or periodic boundaryconditions, are used in this work as a paradigm. The complete analysis that ispresented here can be extended to other two- and three-dimensional dissipativesystems under the assumption of global existence and uniqueness.

In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when 0 ≤ β < 1 2 , the total energy of solutions is not uniformly exponentially stable, but it decays uniformly polynomially to zero. Finally, the results are applied to space semi-discretizations of thermoelastic beam equation in a bounded interval with homogeneous Dirichlet boundary conditions. We consider finite element, spectral element and finite difference semi-discretizations. Finally, we illustrate the mathematical results with several numerical experiments. [ABSTRACT FROM AUTHOR]

We prove the controllability and the existence of the time optimal control for semilinear parabolic equations with the homogenous Neumann boundary condition via multiplicative controls. It is worth pointing out that there is no restriction on the growth of the nonlinearity f(s) with respect to the variable s in the equation, which is a remarkable difference compared to the semilinear parabolic system with additive locally distributed controls. [ABSTRACT FROM AUTHOR]

A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary exponential stabilization where the goal is to present a state feedback law with actuation on only one end of the domain which provide the exponential stability of the closed-loop system. We consider both the Dirichlet and Neumann boundary conditions. The corresponding proposed control law is given in explicit form. Numerical simulations are carried out to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators.

We apply the Postprocessing Galerkin method to a recently introducedcontinuous data assimilation (downscaling) algorithm for obtaining a numericalapproximation of the solution of the two-dimensional Navier-Stokes equationscorresponding to given measurements from a coarse spatial mesh. Under suitableconditions on the relaxation (nudging) parameter, the resolution of the coarsespatial mesh and the resolution of the numerical scheme, we obtain uniform intime estimates for the error between the numerical approximation given by thePostprocessing Galerkin method and the reference solution corresponding to themeasurements. Our results are valid for a large class of interpolant operators,including low Fourier modes and local averages over finite volume elements.Notably, we use here the 2D Navier-Stokes equations as a paradigm, but ourresults apply equally to other evolution equations, such as the Boussinesqsystem of Benard convection and other oceanic and atmospheric circulationmodels.

Analyzing the validity and success of a data assimilation algorithm when somestate variable observations are not available is an important problem inmeteorology and engineering. We present an improved data assimilation algorithmfor recovering the exact full reference solution (i.e. the velocity andtemperature) of the 3D Planetary Geostrophic model, at an exponential rate intime, by employing coarse spatial mesh observations of the temperature alone.This provides, in the case of this paradigm, a rigorous justification to anearlier conjecture of Charney which states that temperature history of theatmosphere, for certain simple atmospheric models, determines all other statevariables.

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtain improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present a few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the convergence analysis of the data assimilation algorithm.

Motivated by the presence of a finite number of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages for dissipative dynamical systems, we present here a continuous data assimilation algorithm for three-dimensional viscous hydrodynamic models. However, to validate the convergence of this algorithm our proofs require the existence of uniform global bounds on the gradients of the solutions of the underlying system in terms of certain combinations of the physical parameters (such as kinematic viscosity, the size of the domain and the forcing term). Therefore our proofs cannot be applied to the three-dimensional Navier-Stokes equations; instead we demonstrate the implementation of this algorithm, for instance, in the context of the three-dimensional Navier-Stokes-α equations. This algorithm consists of introducing a nudging process through a general type of approximation interpolation operator (which is constructed from observational measurements) that synchronizes the large spatial scales of the approximate solutions with those of unknown solutions of the Navier-Stokes-α equations corresponding to these measurements. Our main result provides conditions on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, which is obtained from this collected data, converges to the unknown reference solution over time. These conditions are given in terms of the physical parameters.

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem for optimal control of a viscous generalized θ\theta -type dispersive equation (VG θ\theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG θ\theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

In this study, the new traveling wave solutions resulting from the interaction of the long-short wave system were obtained by using the exp-function method. Two and three dimensional graphs of the obtained solutions were drawn by selecting the appropriate parameters. Density graphs of the solution functions were obtained and the interaction of the waves was observed.

Emerging as a new field, quantum computation has reinvented the fundamentals of Computer Science and knowledge theory in a manner consistent with quantum physics. The fact that quantum computation has superior features and new events than classical computation provides benefits in proving mathematical theories. With advances in technology, the nonlinear partial differential equations are used in almost every area, and many difficulties have been overcome by the solutions of these equations. In particular, the complex solutions of KdV and Burgers equations have been shown to be used in modeling a simple turbulence flow. In this study, Burger-like equation with complex solutions is defined in Hilbert space and solved with an example. In addition, these solutions were analyzed. Thanks to the Quantum Burgers-Like equation, the nonlinear differential equation is solved by linearizing. The pattern changes of time made the result linear. This means that the Quantum Burgers-Like equation can be used to smoothen the sonic processing.

In this article, some new travelling wave solutions of the (3+1) dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation are obtained using the modified exponential function method. When the solution functions obtained are examined, it is seen that functions with periodic functions are obtained. Two and three dimensional graphs of the travelling wave solutions of the BLMP equation are drawn by selecting the appropriate parameters

We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution, and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

The leaderless and leader-following finite-time consensus problems for multi-agent systems (MASs) described by first-order linear hyperbolic partial differential equations (PDEs) are studied. The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs. Finally, two numerical examples are provided to verify the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

We present a computational study of a simple finite-dimensional feedbackcontrol algorithm for stabilizing solutions of infinite-dimensional dissipativeevolution equations such as reaction-diffusion systems, the Navier-Stokesequations and the Kuramoto-Sivashinsky equation. This feedback control schemetakes advantage of the fact that such systems possess finite number ofdetermining parameters or degrees of freedom, namely, finite number ofdetermining Fourier modes, determining nodes, and determining interpolants andprojections. In particular, the feedback control scheme uses finitely many ofsuch observables and controllers that are acting on the coarse spatial scales.We demonstrate our numerical results for the stabilization of the unstable zerosolution of the 1D Chafee-Infante equation and 1D Kuramoto-Sivashinkskyequation. We give rigorous stability analysis for the feedback controlalgorithm and derive sufficient conditions relating the control parameters andmodel parameter values to attune to the control objective.

We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier-Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation interpolation operator exists. Under the assumption that the observational measurements are free of noise, our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data. © 2013 Springer Science+Business Media New York.

We present in this paper a survey of recent results on the controllability of the parabolic system governed by bilinear control. We first discuss the problem of global controllability which corresponds to the question of whether the solution of the system can be driven to a given state at a some finite time by means of a control. We give some results on the global controllability of bilinear and semilinear reaction-diffusion equations. After this we introduce the case of partial controllability with the control acting on a subregion of the domain. Illustrative examples are also provided.

This paper deals with the controllability of the second-grade fluids, a class of non-Newtonian of differential type, on a two-dimensional torus. Using the method of Agrachev and Sarychev (J. Math Fluid Mech., 7(1):108–52 (2005)), Agrachev and Sarychev (Commun Math Phys., 265(3):673–97 (2006)), and of Shirikyan (Commun Math Phys., 266(1):123–51 (2006)), we prove that the system of second-grade fluids is approximately controllable by a finite-dimensional control force. [ABSTRACT FROM AUTHOR]