Sufficient Conditions For Well-posedness Research Articles

A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.

Reduced thin-sandwich equations on manifolds Euclidean at infinity and on closed manifolds: Existence and multiplicity

The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown that the RTSE are well posed in a neighbourhood of umbilical solutions of the constraint equations without conformal Killing fields. In this paper we will analyse such set of equations on manifolds euclidean at infinity in a neighbourhood of asymptotically euclidean (AE) solutions of the ECE. The main conclusion in this direction is that on AE-manifolds admitting a Yamabe positive metric, the solutions of the RTSE parametrize an open subset in the space of solutions of the ECE. Also, we show that in the case of closed manifolds, these equations are well-posed around umbilical solutions of the ECE admitting Killing fields and present some relevant examples. Finally, it will be shown that in the set of umbilical solutions of the vacuum ECE on closed manifolds, the RTSE are generically well-posed.

Boundary-value problems in a layer for evolutionary pseudo-differential equations with integral conditions

Boundary-value problems for evolutionary pseudo-differential equations with an integral condition are studied. Necessary and sufficient conditions of well-posedness are obtained for these problems in the Schwartz spaces. Existence of a well-posed boundary-value problem is proved for each evolutionary pseudo-differential equation.

Well posedness conditions for Bimodal Piecewise Affine Systems

This paper considers well-posedness (the existence and uniqueness of the solutions) of Bimodal Piecewise Affine Systems in Rn. It is assumed that both modes are observable, but only one of the modes is in observable canonical form. This allows the vector field to be discontinuous when the trajectories change mode. Necessary and sufficient conditions for well-posedness are given as a set of algebraic conditions and sign inequalities. It is shown that these conditions induce a joint structure for the system matrices of the two modes. This structure can be used for the classification of well-posed bimodal piecewise affine systems. Furthermore, it is also shown that, under certain conditions, well-posed Bimodal Piecewise Affine Systems in Rn may have one or two equilibrium points or no equilibrium points.

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

AbstractA reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.

Markov selections for the 3D stochastic Navier–Stokes equations

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier-Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.

Well-posedness of initial value problem for Euler equations of inviscid compressible adiabatic fluid

The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local (exact) solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.

On solution concepts and well-posedness of linear relay systems

In this paper we study the well-posedness (existence and uniqueness of solutions) of linear relay systems with respect to two different solution concepts, Filippov solutions and forward solutions. We derive necessary and sufficient conditions for well-posedness in the sense of Filippov of linear systems of relative degree one and two in closed loop with relay feedback. To be precise, uniqueness of Filippov (and also forward) solutions follows in this case if the first non-zero Markov parameter is positive. By means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two. The influence of the Zeno phenomenon (an infinite number of relay switching times in a finite length time interval) on well-posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems, the consequences of the presence of the Zeno behaviour is typical for many other classes of non-smooth and hybrid systems.

Coercivity properties and well-posedness in vector optimization

This paper studies the issue of well-posedness for vector optimization. It is shown that coercivity implies well-posedness without any convexity assumptions on problem data. For convex vector optimization problems, solution sets of such problems are non-convex in general, but they are highly structured. By exploring such structures carefully via convex analysis, we are able to obtain a number of positive results, including a criterion for well-posedness in terms of that of associated scalar problems. In particular we show that a well-known relative interiority condition can be used as a sufficient condition for well-posedness in convex vector optimization.

Structural analysis of robust control systems

In this paper, a necessary and sufficient condition for well-posedness is presented using the inclusion condition of uncertainty, and a set of all the systems corresponding to the uncertainty with which a closed-loop system is well-posed is given. Furthermore, we analyze the topology of the set of systems based on the well-posedness property. We give the number of connected components in the set of uncertainty in the cases of complex matrix uncertainty and 2/spl times/2 real matrix uncertainty.

On Well-Posedness of Relay Systems

In this paper we study the well-posedness (existence and uniqueness of solutions) of linear relay systems with respect to two different solution concepts. We derive necessary and sufficient conditions for well-posedness in the sense of Filippov of linear systems of relative degree one and two in closed loop with relay feedback. For systems of relative degree one and two it is shown that uniqueness of Filippov solutions follows if the first nonzero Markov parameter is positive. By means of an example it is shown that this result is untrue for relative degree larger than two.

THE CAUCHY PROBLEM FOR A DIFFERENTIAL INCLUSION IN BANACH SPACES AND DISTRIBUTION SPACES

We study well-posedness of degenerate Cauchy problems treated as Cauchy problems for a differential inclusion with a multivalued linear operator. Using a new approach to the definition of degenerate integrated semigroups and their generators in a Banach space, we obtain a well-posedness criterion for the problem. Moreover, we consider the Cauchy problem for a differential inclusion in the space of abstract distributions and give necessary and sufficient conditions for well-posedness in the distribution space.

Well-posedness for Optimization Problems with Constraints defined by Variational Inequalities having a unique solution

We introduce various notions of well-posedness for a family of variational inequalities and for an optimization problem with constraints defined by variational inequalities having a unique solution. Then, we give sufficient conditions for well-posedness of these problems and we present an application to an exact penalty method.

Well-posedness of the cauchy problem for complete second-order operator-differential equations

For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|1/2)) and for weak well-posedness of the Cauchy problem in H×H_(|A|+|B|1/2+1). This condition is expressed in terms of location of the joint spectrum of the operators A and B in C2. In terms of location of the spectrum of the operator pencil z2+Az+B in C1, such a condition cannot be written.

On the well-posedness of the initial value problem in elasto-plastodynamics for a linear comparison solid

This work deals with the well-posedness of the linearized initial value problem in solid elastoplastodynamics. By well-posedness we understand existence, uniqueness and continuous dependence of the solution with respect to initial and boundary data. The initial value problem is studied in its most general form using a technique from Kreiss [1]. The cases of elastic and elastic-plastic materials with associated and non-associated plastic flow rules are considered. The analysis is carried under the assumption that the material response remains within the same constitutive cone everywhere (i.e.; exclude loading/unloading subregions). A complete answer to the well-posedness question is given without the requirement of major symmetry of the constitutive tensor. First it is shown that in 2-D the necessary and sufficient condition for well-posedness is the strong ellipticity condition. Up to now this condition had been known only to be a sufficient condition.

On well-posedness of the Papoulis generalized sampling expansion

K.F. Cheung and R.J. Marks (ibid., vol.CAS-32, no.5 p.481-4, 1985) have shown that if at least one of the interpolation functions used in the generalized sampling expansion of A. Papoulis (1977) is not square-integrable, then the problem is ill-posed in the sense that the variance of the reconstruction error is unbounded when noisy samples are used. It is shown that if all the interpolation functions are square-integrable, then the generalized sampling problem is well posed, essentially a converse of the Cheung-Marks result. A useful alternative sufficient condition for well-posedness that depends only on the transfer characteristics of the channels and does not require explicit calculation of the interpolation functions is developed. Several examples illustrating the results are given; in particular, the important nonuniform sampling strategy of J.L. Yen (IRE Trans. Circuits Theory, vol. CT-3, p.251-7, Dec. 1956) using bunched samples is shown to be well posed. >

On a Sufficient Condition for Well-posedness In Gevrey Classes of Some Weakly Hyperbolic Cauchy Problems

On etudie la formulation correcte du probleme de Cauchy pour quelques operateurs faiblement hyperboliques dans les classes de Gevrey

Indefinite Quadratic Cost Optimal Control for Distributed System with Unbounded Control Operator

Based on extensions of linear operator semigroup on Hilbert spaces, the indefinite quadratic optimal control of linear distributed system with unbounded control operator is considered. Necessary and sufficient conditions for wellposedness and solvability are obtained. Closed-loop synthesis of optimal control is proved to be a linear state feedback. The feedback operator function satisfies an integral Riccati equation and has some new properties.

A Characterization of Well-Posed Optimal Control Systems

A constrained optimal regulator problem is considered. Continuous dependence of the optimal control on the desired trajectory (Hadamard well-posedness) or convergence toward the optimal control of any minimizing sequence (Tykhonov well-posedness) are proved when the dynamics are affine (linear plus constant). Dense well-posedness is obtained in the non-affine case. A necessary and sufficient condition for well-posedness for all desired trajectories is shown to be the affine structure of the plant.

Equilibrium solutions of age-specific population dynamics of several species

A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L1 of the dynamics and for existence as well as for stability of equilibrium solutions are given.