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In this paper, we investigate the existence of optimal pairs for optimal control problems with their states constrained pointwise to Riemannian manifolds. For this purpose, by means of the Riemannian geometric tool, we introduce a crucial Cesari-type property, which is an extension of the classical Cesari property (see Definition 3.3, p. 51 in [L. D. Berkovitz, Optimal Control Theory, Appl. Math. Sci. 12, Springer-Verlag, New York, Heidelberg, 1974]) from the setting of Euclidean spaces to that of Riemannian manifolds. Moreover, we show the efficiency of our result by a concrete example. [ABSTRACT FROM AUTHOR]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

In this paper we introduce a general framework for time-inconsistent optimal control problems. We characterize the closed-loop equilibrium strategy in both the integral and pointwise forms with the newly developed methodology. We recover and improve the results of some wellknown models, including the classical optimal control, Bjork, Khapko, and Murgoci [Finance Stoch., 21 (2017), pp. 331-360], He and Jiang, preprint, ssrn:3308274, 2020, and Yong [Math. Control Related Fields, 2 (2012), pp. 271-329] models, and reveal some interesting aspects that appear for the first time in the literature. We illustrate the usefulness of the model and the results by a number of examples in dynamic portfolio selection, including mean-variance with state-dependent risk aversion, investment/consumption with nonexponential discounting, and utility-deviation-risk with coupled terminal state and expected terminal state. [ABSTRACT FROM AUTHOR]

We consider the shape-topological control of a singularly perturbed variational inequality. The geometry-dependent state problem that we address in this paper concerns a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in two dimensions. The corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework. For the reason of asymptotic analysis, singular perturbation theory is applied, resulting in the topological sensitivity of an objective function representing the release rate of the strain energy. In the vicinity of the nonlinear crack, the antiplane strain energy release rate is expressed by means of the mode-III stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions of varying stiffness. The result of shape-topological control is useful either for arrests or rise of crack growth. [ABSTRACT FROM AUTHOR]

We show in this article how perturbative approaches from N. Burq and M. Hitrik [Math. Res. Lett., 14 (2007), pp. 35--47] and the black box strategy from N. Burq and M. Zworski [J. Amer. Math. Soc., 17 (2004), pp. 443--471] allow us to obtain decay rates for Kelvin--Voigt damped wave equations from quite standard resolvent estimates: Carleman estimates or geometric control estimates for Helmoltz equation; Carleman or other resolvent estimates for the Helmoltz equation. Though in this context of Kelvin--Voigt damping, such an approach is unlikely to allow for the optimal results when additional geometric assumptions are considered, it turns out that using this method, we can obtain the usual logarithmic decay which is optimal in general cases. We also present some applications of this approach giving decay rates in some particular geometries (tori). [ABSTRACT FROM AUTHOR]

We consider n × n hyperbolic systems of balance laws in one space dimension @tH(u) + @xF(u) = G(u); t > 0; 0 < x < L, under the assumption that all negative (resp., positive) characteristics are linearly degenerate. We prove the local exact one-sided boundary null controllabil-ity of entropy solutions to this class of systems, which generalizes the corresponding results obtained in [T. Li and L. Yu, J. Math. Pures Appl. (9), 107 (2017), pp. 1{40] from the case without source terms to that with source terms. In order to apply the strategy used in [T. Li and L. Yu, J. Math. Pures Appl. (9), 107 (2017), pp. 1{40], we essentially modify the constructive method by introducing two different kinds of approximate solutions to the system in the forward sense and to the system in the rightward (resp., leftward) sense, respectively, while their limit solutions are equivalent to some extent. [ABSTRACT FROM AUTHOR]

Stabilization to a trajectory for the monodomain equations, a coupled nonlinear PDE-ODE system, is investigated. The results rely on stabilization of linear first-order in time nonautonomous evolution equations combined with stabilizability results for the linearized monodomain equations and a fixed point argument to treat local stabilizability of the nonlinear system. Numerical experiments for feedback stabilization of reentry phenomena are included. [ABSTRACT FROM AUTHOR]

A new method providing necessary and sufficient conditions to test delay-independent stability for general linear time-invariant systems with constant delays is proposed. The method is utilized for single delay and incommensurate multiple delay systems. The proposed method offers an approach to determine the exact boundaries of unknown parameters such as controller gains or system parameters ensuring delay-independent stability, in addition to exhibiting an efficient test for real parameters. The technique is based on nonexistence of unitary complex zeros of an auxiliary characteristic polynomial obtained via extended Kronecker summation. A special feature of the polynomial, i.e., the self-inversive property, is proved and utilized to check its unitary zeros to determine delay-independent stability by an efficient zero location test. The methodology is executed employing simple algebraic operations and inspection of the number of sign variations in the obtained sequence. For the single delay case, the procedure does not require parameter (or frequency) sweeping, equation solving, and pointwise testing even for the determination of the delayindependent stabilizing regions of unknown parameters. In the case of systems with p multiple delays, (p - 2) agent parameters in the range [0, 2π] and one agent parameter in the range [0, π] are swept to determine delay-independent stability without the requirement of solving equations. A graphical projection approach for multiple delays is proposed in the case in which unknown parameters exist. The complete delay-independent stability analysis of a second order PD-controlled system with single delay is presented. Moreover, the method is applied to find the exact delay-independent stabilizing regions of unknown parameters of systems with two and three delays. [ABSTRACT FROM AUTHOR]