Absolute integrability of Mercer kernels is only sufficient for RKHS stability

Reproducing kernel Hilbert spaces (RKHSs) are special Hilbert spaces in one-to-one correspondence with positive definite maps called kernels. They are widely employed in machine learning to reconstruct unknown functions from sparse and noisy data. In the last two decades, a subclass known as stable RKHSs has been also introduced in the setting of linear system identification. Stable RKHSs contain only absolutely integrable impulse responses over the positive real line. Hence, they can be adopted as hypothesis spaces to estimate linear, time-invariant and BIBO stable dynamic systems from input-output data. Necessary and sufficient conditions for RKHS stability are available in the literature and it is known that kernel absolute integrability implies stability. Working in discrete-time, in a recent work we have proved that this latter condition is only sufficient. Working in continuous-time, it is the purpose of this note to prove that the same result holds also for Mercer kernels

On some connections between 2D spectral factorizability and the causal optimal control problem

In this paper the 2D causal optimal control problem is stated and deeply investigated. An explicit equation for the optimal input is derived, even if it appears very difficult to deal with. The 2D spectral factorization problem is introduced, too, and its relationships with the solvability of the previous equation are investigated. The main result consists of a technique for finding the optimal input when spectral factorization can be successfully performed

A note about optimal and suboptimal disturbance rejection

The well-known problem of optimal disturbance re- jection (both in a stochastic framework and in a mean square sense), is addressed by following a Wiener filtering approach. By focusing the analysis to closed loop structures, it is shown that some critical situations may arise, since both the controller causality constraint and the closed loop stability one may prevent the existence of an op- timal controller. When this happens, it is also shown that the mean square infimum value can be approached arbitrarily close by resorting to suitable suboptimal controllers sequences, which agree with both the causality and stability constraints

On some connections between 2D spectral factorizability and the causal optimal control problem

In this paper the 2D causal optimal control problem is stated and deeply investigated. An explicit equation for the optimal input is derived, even if it appears very difficult to deal with. The 2D spectral factorization problem is introduced, too, and its relationships with the solvability of the previous equation are investigated. The main result consists of a technique for finding the optimal input when spectral factorization can be successfully performed

Sliding-Mode Theory Under Feedback Constraints and the Problem of Epidemic Control

One of the most important branches of nonlinear control theory of dynamical systems is the so-called sliding mode. Its aim is the design of a (nonlinear) feedback law that brings and maintains the state trajectory of a dynamic system on a given sliding surface. Here, dynamics become completely independent of the model parameters and can be tuned accordingly to the desired target. In this paper we study this problem when the feedback law is subject to strong structural constraints. In particular, we assume that the control input may take values only over two bounded and disjoint sets. Such sets could be also not perfectly known a priori. An example is a control input allowed to switch only between two values. Under these peculiarities, we derive the necessary and sufficient conditions that guarantee sliding-mode control effectiveness for a class of time-varying continuous time linear systems that includes all the stationary state-space linear models. Our analysis covers several scientific fields. It is only apparently confined to the linear setting and also allows one study an important set of nonlinear models. We describe fundamental examples related to epidemiology where the control input is the level of contact rate among people and the sliding surface permits to control the number of infected. We prove the global convergence of epidemic sliding-mode control schemes applied to two popular dynamical systems used in epidemiology, i.e., SEIR and SAIR, and based on the introduction of severe restrictions like lockdowns. Results obtained in the literature regarding control of many other epidemiological models are also generalized by casting them within a general sliding-mode theory

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