Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity

This paper deals with a control-constrained linear-quadratic optimal control problem governed by the Stokes equations. It is concerned with situations where the gradient of the velocity field is not bounded. The control is discretized by piecewise constant functions. The state and the adjoint state are discretized by finite element schemes that are not necessarily conforming. The approximate control is constructed as projection of the discrete adjoint velocity in the set of admissible controls. It is proved that under certain assumptions on the discretization of state and adjoint state this approximation is of order 2 in L ²(Ω). As first example a prismatic domain with a reentrant edge is considered where the impact of the edge singularity is counteracted by anisotropic mesh grading and where the state and the adjoint state are approximated in the lower order Crouzeix-Raviart finite element space. The second example concerns a nonconvex, plane domain, where the corner singularity is treated by isotropic mesh grading and state and adjoint state can be approximated by a couple of standard element pairs.     

Asymptotics of repeated interaction quantum systems

A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling between S and E. We show that the system, initially in any state close to a reference state, approaches a repeated interaction asymptotic state in the limit of large times. This state is τ-periodic in time and does not depend on the initial state. If the reference state is chosen so that S and E are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.     

Statistical properties of the state obtained by solving a nonlinear multivariate inverse problem

This article takes a statistical approach to solving a multivariate state‐space problem where many data are nonlinearly related to a state vector. The state is unknown and to be predicted, but the problem can be ill posed. A state‐space model quantifies the variability of the physical process (state equation) and of the measurements related to the process (measurement equation). The resulting posterior distribution is then maximized, yielding the predicted state vector. Statistical properties of the predicted state vector, in particular its first two moments with respect to the joint distribution, are approximated using the delta method. These are then applied to the problem of retrieving, from satellite data, a profile of CO₂ values in a column of the atmosphere. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimization of a time-variable,constrained control matrix

The problem posed is to choose, in a optimal manner, a time-variable, bounded, linear transformation defining the velocity of a state point inn-dimensional space in terms of the state. The two-point boundary-value problem which arises from an application of the Pontryagin maximum principle is explicitly solvable; hence, a formula is derived showing that the optimal trajectories in state space are equiangular spirals in two-dimensional subspaces ofR ⁿ and also describing the boundary of the set of attainability. This formula is used to solve the problem of minimal-time transfer between any two given points, and the optimal control is specified both as an open-loop and a closed-loop controller. The solutions to the problem of maximizing a linear payoff function of the final state and of maximizing the angle of rotation of the state vector about the origin are also given.     

Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations

We first study the initial value problem for a general semilinear heat equation. We prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low (n?10) or if the steady state is flat enough at infinity: the solution of the heat equation either becomes unbounded as t approaches the lifespan, or eventually stays above or below another bounded radial steady state, depending on if the initial value is above or below the first steady state; moreover, the second steady state must be a constant if n?10.Using this instability result, we then prove that every nonconstant radial steady state of the generalized Fisher equation is a hair-trigger for two kinds of dynamical behavior: extinction and spreading. We also prove more criteria on initial values for these types of behavior. Similar results for a reaction-diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained.     

Orthogonal polynomials with a resolvent-type generating function

The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

     

Optimal production correction and maintenance policy for imperfect process

This study considers imperfect production processes that require production correction and maintenance. Two states of the production process are performed, namely: the type I state (out-of-control state) and the type II state (in-control state). At the beginning of the production of the each renewal cycle, the state of the process is assumed not always to be restored to “in-control”. The type I state involves the adjustment of the production mechanism, whereas the type II state does not. Production correction is either imperfect; worsening a production system, or perfect, returning it to “in-control”. After N + 1 type I states, the operating system must be maintained and returned to the beginning condition. The mean loss cost due to reproduction through production correction per the total expected cost until the N + 1 type I states are entered successively is determined. The existence of a unique and finite optimal N for an imperfect process under certain reasonable conditions is shown. A numerical example is presented.     

On factorial states of operator algebras

Results for the factorial state space of a C¹-algebra A which are analogous to results of 11., 12., 572–612),Tomiyama and Takesaki (Tohôku Math. J. (2) 13 (1961), 498–523) for the pure state space. It is shown that A is prime if and only if the (type I) factorial states are dense in the state space. It follows that every factorial state is a w¹-limit of type I factorial states. The factorial state space of a von Neumann algebra is determined, and it is shown that if A is unital and acts non-degenerately on a Hilbert space then the factorial state space of the generated von Neumann algebra restricts precisely to the factorial state space of A. It is shown that the set of factorial states is w¹-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.     

Der plastische Grenzzustand in der schiefen ebenen Erd- oder Schneeschicht

Summary On the basis of recent articles byD. C. Drucker, W. Prager, R. T. Shield andH. Ziegler, the author establishes general relations for the state of plane flow in an idealized soil. A simple geometric interpretation of the relation between the stresses and the strain rates is given. The condition that the dissipation rate associated with an admissible state of motion cannot be negative is transformed into a new criterion, which in many cases facilitates the decision as to whether a state of motion is compatible with a prescribed state of stress or not. As an example, a study is made of the plastic state of an inclined plane layer. Particular emphasis is paid to a comparison of layers of equal inclination but of different cohesion and internal friction. Layers of variable density are also considered.     

State subsidy and moral hazard in corporate financing

This paper investigates the impact of state subsidy on the behavior of the entrepreneur under asymmetric information. Several authors formulated concerns about state intervention as it can aggravate moral hazard in corporate financing. In the seminal paper of Holmström and Tirole (Q J Econ 112(3):663–691, 1997) a two-player moral hazard model is presented with an entrepreneur initiating a risky scalable project and a private investor (e.g. bank or venture capitalist) providing outside financing. The novelty of our research is that this basic moral hazard model is extended to the case of positive externalities and to three players by introducing the state subsidizing the project. It is shown that in the optimum, state subsidy does not harm, but improves the incentives of the entrepreneur to make efforts for the success of the project; hence in effect state intervention reduces moral hazard. Consequently, state subsidy increases social welfare which is defined as the sum of private and public net benefits. Also, the exact form of the state subsidy (ex-ante/ex-post, conditional/unconditional, refundable/nonrefundable) is irrelevant in respect of the optimal size and the total welfare effect of the project. Moreover, in case of nonrefundable subsidies state does not crowd out private investors; but on the contrary, by providing additional capital it boosts private financing. These results are mainly due to the special mechanism imbedded in our model by which the private investor is able to transform even the badly designed state subsidies into a success fee which is optimal from the incentive point of view.     

Inspection scheduling for imperfect production processes under free repair warranty contract

The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.     

State-Feedback Stabilization of Well-Posed Linear Systems

A finite-dimensional linear time-invariant system is output-stabilizable if and only if it satisfies the finite cost condition, i.e., if for each initial state there exists at least one L² input that produces an L² output. It is exponentially stabilizable if and only if for each initial state there exists at least one L² input that produces an L² state trajectory. We extend these results to well-posed linear systems with infinite-dimensional input, state and output spaces. Our main contribution is the fact that the stabilizing state feedback is well posed, i.e., the map from an exogenous input (or disturbance) to the feedback, state and output signals is continuous in L_loc² in both open-loop and closed-loop settings. The state feedback can be chosen in such a way that it also stabilizes the I/O map and induces a (quasi) right coprime factorization of the original transfer function. The solution of the LQR problem has these properties.     

A fluid model with upward jumps at the boundary

We consider a single buffer fluid system in which the instantaneous rate of change of the fluid is determined by the current state of a background stochastic process called “environment”. When the fluid level hits zero, it instantaneously jumps to a predetermined positive level q. At the jump epoch the environment state can undergo an instantaneous transition. Between two consecutive jumps of the fluid level the environment process behaves like a continuous time Markov chain (CTMC) with finite state space. We develop methods to compute the limiting distribution of the bivariate process (buffer level, environment state). We also study a special case where the environment state does not change when the fluid level jumps. In this case we present a stochastic decomposition property which says that in steady state the buffer content is the sum of two independent random variables: one is uniform over [0,q], and the other is the steady-state buffer content in a standard fluid model without jumps.      

Well-Posed State/Signal Systems in Continuous Time

We introduce a new class of linear systems, the L ^p -well-posed state/signal systems in continuous time, we establish the foundations of their theory and we develop some tools for their study. The principal feature of a state/signal system is that the external signals of the system are not a priori divided into inputs and outputs. We relate state/signal systems to the better-known class of well-posed input/state/output systems, showing that state/signal systems are more flexible than input/state/output systems but still have enough structure to provide a meaningful theory. We also give some examples which point to possibilities for further study.     

A method for analysis of linear dynamic systems driven by stationary non-Gaussian noise with applications to turbulence-induced random vibration

A method is developed for approximating the properties of the state of a linear dynamic system driven by a broad class of non-Gaussian noise, namely, by polynomials of filtered Gaussian processes. The method involves four steps. First, the mean and correlation functions of the state of the system are calculated from those of the input noise. Second, higher order moments of the state are calculated based on Itô’s formula for continuous semimartingales. It is shown that equations governing these moments are closed, so that moment of any order of the state can be calculated exactly. Third, a conceptually simple technique, which resembles the Galerkin method for solving differential equations, is proposed for constructing approximations for the marginal distribution of the state from its moments. Fourth, translation models are calibrated to representations of the marginal distributions of the state as well as its second moment properties. The resulting models can then be utilized to estimate properties of the state, such as the mean rate at which the state exits a safe set. The implementation of the proposed method is demonstrated by numerous examples, including the turbulence-induced random vibration of a flexible plate.     

Guaranteed Nonlinear State Estimator for Cooperative Systems

This paper is about state estimation for continuous-time nonlinear models, in a context where all uncertain variables can be bounded. More precisely, cooperative models are considered, i.e., models that satisfy some constraints on the signs of the entries of the Jacobian of their dynamic equation. In this context, interval observers and a guaranteed recursive state estimation algorithm are combined to enclose the state at any given instant of time in a subpaving. The approach is illustrated on the state estimation of a waste-water treatment process.     

Moments of Change

There is a strong intuition that for a change to occur, there must be a moment at which the change is taking place. It will be demonstrated that there are no such moments of change, since no state the changing thing could be in at any moment would suffice to make that moment a moment of change. A moment in which the changing thing is simply in the state changed from or the state changed to cannot be the moment of change, since these states are respectively before and after the change; moreover, to select one of these moments over the other as the moment of change would be arbitrary. A moment in which the changing thing is neither in the state changed from nor in the state changed to cannot be the moment of change, since there are changes for which it is impossible for something to be in neither state. Finally, the moment of change cannot be a moment in which the changing thing is in both the state changed from and the state changed to, as suggested by Graham Priest and others. Even if, like proponents of this view, we are willing to accept the contradictions that the account entails, it is demonstrated that on such a model, every change would require an infinite number of other changes, every change would take an infinite amount of time, and some changes would occur without occurring at any time. Further, the model is grossly counterintuitive, with the exact nature of the counterintuitive element depending on what model of time and space one endorses. Finally, it is demonstrated that this model is incompatible with the Leibniz Continuity Condition.     

Convergent expansions in non-relativistic qed: Analyticity of the ground state

We consider the ground state of an atom in the framework of non-relativistic qed. We show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential, under the assumption that the atomic Hamiltonian has a non-degenerate ground state. Moreover, we show that the corresponding expansion coefficients are precisely the coefficients of the associated Raleigh-Schrödinger series. As a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the Rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant α, with α dependent coefficients which are finite for α?0.