Abstract. Optimal and robust control theories are used to determine effective, estimator-based feedback control rules for laminar plane channel flows that effectively stabilize linearly unstable flow perturbations at Re=10 000 and linearly stable flow perturbations, characterized by mechanisms for very large disturbance amplification, at Re=5000. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control, and the flow measurement is derived from the wall skin friction. The control objective, beyond simply stabilizing any unstable eigenvalues (which is relatively easy to accomplish), is to minimize the energy of the flow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance amplification are present, small-amplitude external disturbance forcing may excite flow perturbations with sufficiently large amplitude to induce nonlinear flow instability. The control algorithms used in the present work account for system disturbances and measurement noise in a rigorous fashion by application of modern linear control techniques to the discretized linear stability problem. The disturbances are accounted for both as uncorrelated white Gaussian processes ([script H]2 or ‘optimal’ control) and as finite ‘worst case’ inputs which are maximally detrimental to the control objective ([script H][infty infinity] or ‘robust’ control). Root loci and transient energy growth analyses are shown to be inadequate measures to characterize overall system performance. Instead, appropriately defined transfer function norms are used to characterize all systems considered in a consistent and relevant manner. In order to make a parametric study tractable in this high-dimensional system, a convenient new scaling to the estimation problem is introduced such that three scalar parameters {[gamma], [alpha], [script l]} may be individually adjusted to achieve desired closed-loop characteristics of the resulting systems. These scalar parameters may be intuitively explained, and are defined such that the resulting control equations retain the natural dual structure between the control parameter, [script l], and the estimation parameter, [alpha]. The performance of the present systems with respect to these parameters is thoroughly investigated, and comparisons are made to simple proportional schemes where appropriate.
Abstract. The application of optimal control theory to complex problems in fluid mechanics has proven to be quite effective when complete state information from high-resolution numerical simulations is available [P. Moin, T.R. Bewley, Appl. Mech. Rev., Part 2 47 (6) (1994) S3-S13; T.R. Bewley, P. Moin, R. Temam, J. Fluid Mech. (1999), submitted for publication]. In this approach, an iterative optimization algorithm based on the repeated computation of an adjoint field is used to optimize the controls for finite-horizon nonlinear flow problems [F. Abergel, R. Temam, Theoret. Comput. Fluid Dyn. 1 (1990) 303-325]. In order to extend this infinite-dimensional optimization approach to control externally disturbed flows in which the controls must be determined based on limited noisy flow measurements alone, it is necessary that the controls computed be insensitive to both state disturbances and measurement noise. For this reason, robust control theory, a generalization of optimal control theory, has been examined as a technique by which effective control algorithms which are insensitive to a broad class of external disturbances may be developed for a wide variety of infinite-dimensional linear and nonlinear problems in fluid mechanics. An aim of the present paper is to put such algorithms into a rigorous mathematical framework, for it cannot be assumed at the outset that a solution to the infinite-dimensional robust control problem even exists. In this paper, conditions on the initial data, the parameters in the cost functional, and the regularity of the problem are established such that existence and uniqueness of the solution to the robust control problem can be proven. Both linear and nonlinear problems are treated, and the 2D and 3D nonlinear cases are treated separately in order to get the best possible estimates. Several generalizations are discussed and an appropriate numerical method is proposed.
Abstract. We consider the null controllability problem for the semilinear heat equation with nonlinearities involving gradient terms in an unbounded domain [OHgr] of RN with Dirichlet boundary conditions. The control is assumed to be distributed along a subdomain [ohgr] such that the uncontrolled region [OHgr]\\[ohgr] is bounded. Using Carleman inequalities, we prove first the null controllability of the linearized equation. Then, by a fixed-point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is C1 and globally Lipschitz, the system is null controllable.
Abstract. For boundary or distributed controls, we get an approximate controllability result for the Navier-Stokes equations in dimension 2 in the case where the fluid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.
Abstract. We describe the large time behavior of solutions of the convection-diffusion equation where dN and a=a (x) is a symmetric periodic matrix satisfying suitable ellipticity assumptions. We also assume that aW1, (N). First, we consider the linear problem (d=0) and prove that the large time behavior of solutions is given by the fundamental solution of the diffusion equation with aah where ah is the homogenized matrix. In the nonlinear case, when q=1+, we prove that the large time behavior of solutions with initial data in L1(N) is given by a uniparametric family of self-similar solutions of the convection-diffusion equation with constant homogenized diffusion aah. When q>1+, we prove that the large time behavior of solutions is given by the fundamental solution of the linear-diffusion equation with aah
Abstract. We prove exact controllability for Maxwell's system with variable coefficients in a bounded domain by a current flux in the boundary. The proof relies on a duality argument which reduces the proof of exact controllability to the proof of continuous observability for the homogeneous adjoint system. There is no geometric restriction imposed on the domain.
Abstract. Using nonlinear programming theory in Banach spaces we derive a version of Pontryagin's maximum principle that can be applied to distributed parameter systems under control and state constrains. The results are applied to fluid mechanics and combustion problems.
Abstract. This paper is concerned with the null controllability of systems governed by semilinear parabolic equations. The control is exerted either on a small subdomain or on a portion of the boundary. We prove that the system is null controllable when the nonlinear term f(s) grows slower than s . log|s| as |s| tends to infinity.
Abstract. This paper presents some known results on the approximate and null controllability of the Navier--Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions.
Abstract. We study optimal boundary control problems for the two-dimensional Navier--Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier--Stokes equations; namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier--Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.
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In this paper we study Maxwell's equations with a thermal effect. This system models an induction heating process where the electric conductivity
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We prove analytic criteria for the existence of finite-time attracting and repelling material surfaces and lines in three-dimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finite-time Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from three-dimensional velocity data. We illustrate the results on steady and unsteady ABC-type flows.
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We study the transport of particles in a general, two-dimensional, incompressible flow in the presence of a transient eddy, i.e., a bounded set of closed streamlines with a finite time of existence. Using quantities obtained from Eulerian observations, we provide explicit conditions for the existence of a hyperbolic structure in the flow, which induces mixing between the eddy and its environment. Our results can be used directly to study finite-time transport in numerically or experimentally generated vector fields with general time-dependence.
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The long-time behavior of solutions for an optimal distributed control problem associated with the Navier--Stokes equations is studied. First, a linear feedback solution for the Navier--Stokes equations is constructed; this feedback solution possesses decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of the Navier--Stokes equations are derived. Next, the existence of a solution for the optimal control problem is proved. Finally, the long-time decay properties for the optimal solutions are established.
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We study the local exact controllability problem for the Navier-Stokes equations that describe an incompressible fluid flow in a bounded domain with control distributed in an arbitrary fixed subdomain. The result that we obtain in this paper is as follows. Suppose that we have a given stationary point of the Navier-Stokes equations and our initial condition is sufficiently close to it. Then there exists a locally distributed control such that in a given moment of time the solution of the Navier-Stokes equations coincides with this stationary point.
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We consider an initial and boundary value problem for the one and two dimensional wave equation with nonlinear damping concentrated on an interior point and respectively on an interior curve. In the two dimensional case our main result asserts that generically (i.e., for almost all interior curves) the solutions decay to zero in the energy space. When the domain is strictly convex we show that, whatever the interior curve is, the decay is not uniform. We generalize in this way results known in one space dimension. Our main improvement of existing one-dimensional results consists in giving sharp decay rates, provided that the initial data are regular and the damping term is linear. A crucial intermediate step is the proof of a generalization of Ingham's inequality on nonharmonic Fourier series.
Abstract.
Coherent structures (CS) near the wall (i.e. y+ [less-than-or-eq, slant] 60) in a numerically simulated turbulent channel flow are educed using a conditional sampling scheme which extracts the entire extent of dominant vortical structures. Such structures are detected from the instantaneous flow field using our newly developed vortex definition (Jeong & Hussain 1995) – a region of negative [lambda]2, the second largest eigenvalue of the tensor SikSkj + [Omega]ik[Omega]kj – which accurately captures the structure details (unlike velocity-, vorticity- or pressure-based eduction). Extensive testing has shown that [lambda]2 correctly captures vortical structures, even in the presence of the strong shear occurring near the wall of a boundary layer. We have shown that the dominant near-wall educed (i.e. ensemble averaged after proper alignment) CS are highly elongated quasi-streamwise vortices; the CS are inclined 9° in the vertical (x, y)-plane and tilted ±4° in the horizontal (x, z)-plane. The vortices of alternating sign overlap in x as a staggered array; there is no indication near the wall of hairpin vortices, not only in the educed data but also in instantaneous fields. Our model of the CS array reproduces nearly all experimentally observed events reported in the literature, such as VITA, Reynolds stress distribution, wall pressure variation, elongated low-speed streaks, spanwise shear, etc. In particular, a phase difference (in space) between streamwise and normal velocity fluctuations created by CS advection causes Q4 (‘sweep’) events to dominate Q2 (‘ejection’) and also creates counter-gradient Reynolds stresses (such as Q1 and Q3 events) above and below the CS. We also show that these effects are adequately modelled by half of a Batchelor's dipole embedded in (and decoupled from) a background shear U(y). The CS tilting (in the (x, z)-plane) is found to be responsible for sustaining CS through redistribution of streamwise turbulent kinetic energy to normal and spanwise components via coherent pressure–strain effects.
Abstract.
Coherent structures (CS) near the wall (i.e. y+ [less-than-or-eq, slant] 60) in a numerically simulated turbulent channel flow are educed using a conditional sampling scheme which extracts the entire extent of dominant vortical structures. Such structures are detected from the instantaneous flow field using our newly developed vortex definition (Jeong & Hussain 1995) – a region of negative [lambda]2, the second largest eigenvalue of the tensor SikSkj + [Omega]ik[Omega]kj – which accurately captures the structure details (unlike velocity-, vorticity- or pressure-based eduction). Extensive testing has shown that [lambda]2 correctly captures vortical structures, even in the presence of the strong shear occurring near the wall of a boundary layer. We have shown that the dominant near-wall educed (i.e. ensemble averaged after proper alignment) CS are highly elongated quasi-streamwise vortices; the CS are inclined 9° in the vertical (x, y)-plane and tilted ±4° in the horizontal (x, z)-plane. The vortices of alternating sign overlap in x as a staggered array; there is no indication near the wall of hairpin vortices, not only in the educed data but also in instantaneous fields. Our model of the CS array reproduces nearly all experimentally observed events reported in the literature, such as VITA, Reynolds stress distribution, wall pressure variation, elongated low-speed streaks, spanwise shear, etc. In particular, a phase difference (in space) between streamwise and normal velocity fluctuations created by CS advection causes Q4 (‘sweep’) events to dominate Q2 (‘ejection’) and also creates counter-gradient Reynolds stresses (such as Q1 and Q3 events) above and below the CS. We also show that these effects are adequately modelled by half of a Batchelor's dipole embedded in (and decoupled from) a background shear U(y). The CS tilting (in the (x, z)-plane) is found to be responsible for sustaining CS through redistribution of streamwise turbulent kinetic energy to normal and spanwise components via coherent pressure–strain effects.
Abstract.
This paper considers transmission problem for the system of electromagneto-elasticity having piecewise constant coefficients in a bounded domain. The result on exact boundary controllability is obtained provided the interfaces, where the coefficients have a jump discontinuity, are all star-shaped with respect to one and the same point and the coefficients satisfy a certain monotonicity conditions.
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We consider a transmission problem for Maxwell's equations with a dissipative boundary condition of memory type. Under suitable geometric conditions imposed on the domain and the interfaces where the coefficients are allow to have a jump discontinuity, results on uniform stabilization are established.
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We consider a linear system of thermoelasticity in a compact, Cinfin, n-dimensional connected Riemannian manifold. This system consists of a wave equation coupled to a heat equation. When the boundary of the manifold is non-empty, Dirichlet boundary conditions are considered. We study the controllability properties of this system when the control acts in the hyperbolic equation (and not in the parabolic one) and has its support restricted to an open subset of the manifold. We show that, if the control time and the support of the control satisfy the geometric control condition for the wave equation, this system of thermoelasticity is null-controllable. More precisely, any finite-energy solution can be driven to zero at the control time. An analogous result is proved when the control acts on the parabolic equation. Finally, when the manifold has no boundary, the null-controllability of the linear system of three-dimensional thermoelastic ity is proved.
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In this paper we consider an abstract linear system with perturbation of the form
$$ \frac{dy}{dt}= Ay + \varepsilon By $$
on a Hilbert space ${\cal H}$, where A is skew-adjoint, B is bounded, and $\varepsilon$ is a positive parameter. Motivated by a work of Freitas and Zuazua on the one-dimensional wave equation with indefinite viscous damping [P. Freitas and E. Zuazua, J. Differential Equations, 132 (1996), pp. 338--352], we obtain a sufficient condition for exponential stability of the above system when B is not a dissipative operator. We also obtain a Hautus-type criterion for exact controllability of system (A, G), where G is a bounded linear operator from another Hilbert space to ${\cal H}$. Our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping, as well as the exponential stabilization of the elastic systems with noncolocated observation and control.
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In this paper we prove that the null controllability property of the heat equation may be obtained as limit of the exact controllability properties of singularly perturbed damped wave equations. We impose Dirichlet, homogeneous boundary conditions. The control is supported in a neighborhood of a subset of the boundary that satisfies the classical requirements to apply multiplier techniques. The proof needs an iterative argument that allows to treat separately the low and high frequencies and to make use of the dissipativity of the systems under consideration. This is combined with sharp observability estimates on the eigenfunctions of the Laplacian due to G. Lebeau and L. Robbiano, and global Carleman estimates.
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We examine the question of control of Maxwell's equations in a heterogeneous medium with a nonsmooth boundary by means of control currents on the boundary of that medium. This requires the introduction and analysis of some functions spaces. Some energy estimates are established which allow us to get the control results owing to the Hilbert uniqueness method. We finally give an application to an inverse source problem.
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We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.
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This paper deals with Maxwell's equations coupled with a nonlinear heat equation. The system models an induction heating process for a conductive material in which the electrical conductivity strongly depends on the temperature. It is shown that the evolution system has a global weak solution if the electrical conductivity is bounded. For the case of one space dimension, the existence of a global classical solution is established. Moreover, for a quasi-stationary state field it is proved that the temperature will blow up in finite time if the electric conductivity satisfies certain growth conditions.
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In this paper we obtain two exact internal controllability results of Maxwell's equations in a general region by using multiplier techniques. The first one is exact controllability in a short time, in which we obtain the ``optimal" (observability) estimates when the location and the shape of the controller is fixed. What happens if we allow the controller to change? Under some conditions, we show that by doing that the system can be exactly controllable within any given time duration, which is our second exact controllability result.
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