Uniform convergence of operator semigroups without time regularity

When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of $$C_0$$ C 0 -semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on $${\mathbb {R}}^d$$ R d , the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of $$C_0$$ C 0 -semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations—without any time regularity assumptions—by adapting the concept of the “semigroup at infinity”, recently introduced by M. Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e. powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.

Functional observer-based feedback controller for ball balancing table

The two degree of freedom ball balancing table (BBT) is a well-known didactic tool used to evaluate the effectiveness and performances of many control algorithms for dynamic systems. The present paper proposes to control the ball position of the BBT system via a linear feedback controller based on a functional observer. The parameters of the linear functional observer are determined by applying the direct method which requires neither a Sylvester equation resolution nor canonical transformations. The use of a digital controller has motivated the elaboration of the equations in the discrete time case. In this work, the BBT is tested in real-time to evaluate the proposed controller performances when stabilizing a ball on a reference point. This paper is a continuity of the previous work [12], in which only simulation results have been carried out.

$H_\infty$ State Estimation of Delayed Recurrent Memristive Neural Networks: continuous-time case and discrete-time case

This paper investigates the problem of $H_\infty$ state estimation of delayed recurrent memristive neural networks (DRMNNs) with both continuous-time and discrete-time cases. By utilizing Lyapunov-Krasovskii functional (LKF) and linear matrix inequalities (LMIs), two criterions are provided to guarantee the asymptotically stable of the estimation error systems with a $H_\infty$ performance. The connection weight parameters of DRMNNs are dealed with logical switching signals, which greatly reduces the computational complexity. The given conditions can be easily checked by solving LMIs, the obtained theoretical results are supported demonstrated by two numerical examples.

Demographic inference from multiple whole genomes using a particle filter for continuous Markov jump processes

Demographic events shape a population’s genetic diversity, a process described by the coalescent-with-recombination model that relates demography and genetics by an unobserved sequence of genealogies along the genome. As the space of genealogies over genomes is large and complex, inference under this model is challenging. Formulating the coalescent-with-recombination model as a continuous-time and -space Markov jump process, we develop a particle filter for such processes, and use waypoints that under appropriate conditions allow the problem to be reduced to the discrete-time case. To improve inference, we generalise the Auxiliary Particle Filter for discrete-time models, and use Variational Bayes to model the uncertainty in parameter estimates for rare events, avoiding biases seen with Expectation Maximization. Using real and simulated genomes, we show that past population sizes can be accurately inferred over a larger range of epochs than was previously possible, opening the possibility of jointly analyzing multiple genomes under complex demographic models. Code is available at https://github.com/luntergroup/smcsmc.