A necessary condition for dynamic equivalence

International audienceIf two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different dimensions, the one of higher dimension must ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a Lie-Bäcklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is assumed to be "trivial" (or linear controllable)

Singular nonlinear H∞ optimal control problem

The theory of nonlinear H∞ of optimal control for affine nonlinear systems is extended to the more general context of singular H∞ optimal control of nonlinear systems using ideas from the linear H∞ theory. Our approach yields under certain assumptions a necessary and sufficient condition for solvability of the state feedback singular H∞ control problem. The resulting state feedback is then used to construct a dynamic compensator solving the nonlinear output feedback H∞ control problem by applying the certainty equivalence principle

The pitfalls of deciding whether a quantum channel is (conjugate) degradable and how to avoid them

To decide whether a quantum channel is degradable is relatively easy: one has to find at least one example of a degrading quantum channel. But in general, no conclusive criterion exists to show the opposite. Using elementary methods we derive a necessary and sufficient condition to decide under what circumstances the conclusion is unambiguous. The findings lead to an extension of the antidegradability region for qubit and qutrit transpose depolarizing channels. In the qubit case we reproduce the known results for the class of qubit depolarizing channels (due to their equivalence). One of the consequences is that the optimal qubit and qutrit asymmetric cloners possess a single-letter quantum capacity formula. We also investigate the ramifications of the criterion for the search of exclusively conjugate degradable channels.Comment: v2: Full rank assumption added to the main theorem; to appear in Open Systems & Information Dynamic

Dynamic Mechanism Design: Incentive Compatibility, Profit Maximization and Information Disclosure

This paper examines the problem of how to design incentive-compatible mechanisms in environments in which the agents' private information evolves stochastically over time and in which decisions have to be made in each period. The environments we consider are fairly general in that the agents' types are allowed to evolve in a non-Markov way, decisions are allowed to affect the type distributions and payoffs are not restricted to be separable over time. Our first result is the characterization of a dynamic payoff formula that describes the evolution of the agents' equilibrium payoffs in an incentive-compatible mechanism. The formula summarizes all local first-order conditions taking into account how current information affects the dynamics of expected payoffs. The formula generalizes the familiar envelope condition from static mechanism design: the key difference is that a variation in the current types now impacts payoffs in all subsequent periods both directly and through the effect on the distributions of future types. First, we identify assumptions on the primitive environment that guarantee that our dynamic payoff formula is a necessary condition for incentive compatibility. Next, we specialize this formula to quasi-linear environments and show how it permits one to establish a dynamic "revenue-equivalence" result and to construct a formula for dynamic virtual surplus which is instrumental for the design of optimal mechanisms. We then turn to the characterization of sufficient conditions for incentive compatibility. Lastly, we show how our results can be put to work in a variety of applications that include the design of profit-maximizing dynamic auctions with AR(k) values and the provision of experience goods.dynamic mechanisms, asymmetric information, stochastic processes, incentives

Optimal Dynamic Nonlinear Income Taxes with No Commitment

We wish to study optimal dynamic nonlinear income taxes. Do real world taxes share some of their features? What policy prescriptions can be made? We study a two period model, where the consumers and government each have separate budget constraints in the two periods, so income cannot be transferred between periods. Labor supply in both periods is chosen by the consumers. The government has memory, so taxes in the first period are a function of first period labor income, whereas taxes in the second period are a function of both first and second period labor income. The government cannot commit to future taxes. Time consistency is thus imposed as a requirement. The main results of the paper show that time consistent incentive compatible two period taxes involve separation of types in the first period and a differentiated lump sum tax in the second period, provided that the discount rate is high or utility is separable between labor and consumption. In the natural extension of the Diamond (1998) model with quasi-linear utility functions to two periods, an equivalence of dynamic and static optimal taxes is demonstrated, and a necessary condition for the top marginal tax rate on first period income is found.Optimal Income Taxation; Time Consistency; Incentive Compatibility; Sequential Information Revelation; Optimal Dynamic Taxation

Dynamic Mechanism Design: A Myersonian Approach

We study mechanism design in dynamic quasilinear environments where private information arrives over time and decisions are made over multiple periods. We make three contributions. First, we provide a necessary condition for incentive compatibility that takes the form of an envelope formula for the derivative of an agent's equilibrium expected payoff with respect to his current type. It combines the familiar marginal effect of types on payoffs with novel marginal effects of the current type on future ones that are captured by “impulse response functions.” The formula yields an expression for dynamic virtual surplus that is instrumental to the design of optimal mechanisms and to the study of distortions under such mechanisms. Second, we characterize the transfers that satisfy the envelope formula and establish a sense in which they are pinned down by the allocation rule (“revenue equivalence”). Third, we characterize perfect Bayesian equilibrium-implementable allocation rules in Markov environments, which yields tractable sufficient conditions that facilitate novel applications. We illustrate the results by applying them to the design of optimal mechanisms for the sale of experience goods (“bandit auctions”)

Dynamic and Thermodynamic Stability of Charged Perfect Fluid Stars

We perform a thorough analysis of the dynamic and thermodynamic stability for the charged perfect fluid star by applying the Wald formalism to the Lagrangian formulation of Einstein-Maxwell-charged fluid system. As a result, we find that neither the presence of the additional electromagnetic field nor the Lorentz force experienced by the charged fluid makes any obstruction to the key steps towards the previous results obtained for the neutral perfect fluid star. Therefore, the criterion for the dynamic stability of our charged star in dynamic equilibrium within the symplectic complement of the trivial perturbaions with the ADM $3$-momentum unchanged is given by the non-negativity of the canonical energy associated with the timelike Killing field, where it is further shown for both non-axisymmetric and axisymmetric perturbations that the dynamic stability against these restricted perturbations also implies the dynamic stability against more generic perturbations. On the other hand, the necessary condition for the thermodynamic stability of our charged star in thermodynamic equilibrium is given by the positivity of the canonical energy of all the linear on-shell perturbations with the ADM angular momentum unchanged in the comoving frame, which is equivalent to the positivity of the canonical energy associated with the timelike Killing field when restricted onto the axisymmetric perturbations. As a by-product, we further establish the equivalence of the dynamic and thermodynamic stability with respect to the spherically symmetric perturbations of the static, spherically symmetric isentropic charged star.Comment: 20 pages, 1 figur