Ai Alina Doban

On stability analysis of discrete-time homogeneous dynamics

This paper considers the problem of stability analysis of discrete-time dynamics that are positively homogeneous of degree one. An example of a homogeneous and even continuous dynamics that is globally exponentially stable and that does not admit any λ-contractive proper C-set is presented. This motivates us to propose a natural generalization of this concept, namely, (k, λ)-contractive proper C-sets. It is proven that this simple generalization yields a non-conservative Lyapunov-type tool for stability analysis of homogeneous dynamics, namely, sublinear finite-time Lyapunov functions. Moreover, scalable and non-conservative stability tests are established for relevant classes of homogeneous dynamics.

Domain of attraction computation for tumor dynamics

In this paper we propose the use of rational Lyapunov functions to estimate the domain of attraction of the tumor dormancy equilibrium of immune cells-malignant cells interaction dynamics. A procedure for computing rational Lyapunov functions is worked out, with focus on obtaining a meaningful domain of attraction for the considered tumor dynamics. A valid, nonconvex domain of attraction was obtained, which is consistent with non-trivial tumor evolutions from real-life.

On infinity norms as Lyapunov functions for continuous-time dynamical systems

This paper considers the synthesis of polyhedral Lyapunov functions for continuous-time dynamical systems. A proper conic partition of the state-space is employed to construct a finite set of linear inequalities in the elements of the Lyapunov weight matrix. For dynamics described by linear and polytopic differential inclusions, it is proven that the feasibility of the derived set of linear inequalities is necessary and sufficient for the existence of an infinity norm Lyapunov function. Furthermore, it is shown that the developed solution naturally applies to relevant classes of continuous-time nonlinear systems. An extension to non-symmetric polyhedral Lyapunov functions is also presented.

A switched systems approach to cancer therapy

In this paper, we propose a systematic cancer therapy strategy, which is based on switching between successive parameter dependent domains of attraction. More specifically, we address the problem of steering a stable invasive tumor to tumor dormancy. A predator-prey model from the literature is considered for describing the tumor-normal cells interaction. For computing the domain of attraction of an equilibrium of interest, maximal rational Lyapunov functions are employed, which can be systematically computed for nonlinear systems. The proposed procedure confirms observations from medical practice and provides a useful tool for cancer therapy design and testing.

Computation of Lyapunov Functions for Nonlinear Differential Equations via a Massera-Type Construction

An approach for computing Lyapunov functions for nonlinear continuous-time differential equations is developed via a new, Massera-type construction. This construction is enabled by imposing a finite-time criterion on the integrated function. By means of this approach, we relax the assumptions of exponential stability on the systems equilibrium, while still allowing integration over a finite time interval. The resulting Lyapunov function can be computed based on any

On constrained stabilization of discrete-time linear systems

\mathcal {K}_{\infty}

A switching control law approach for cancer immunotherapy of an evolutionary tumor growth model.

-function of the norm of the solution of the system. In addition, we show how the developed converse theorem can be used to construct an estimate of the domain of attraction. Finally, a range of examples from the literature and biological applications, such as the genetic toggle switch, the repressilator, and the HPA axis, are worked out to demonstrate the efficiency and improvement in computation of the proposed approach.

Feedback stabilization via rational control Lyapunov functions

The concepts of controlled (k, λ)-contractive sets and set-induced finite-time control Lyapunov functions are introduced in this paper. These tools are then employed to derive new synthesis methods for constrained stabilization of linear systems. Two classes of state-feedback control strategies are proposed, namely, periodic conewise linear control laws and periodic vertex-interpolation control laws. The benefits of these synthesis methods are demonstrated for the constrained stabilization of a DC-DC buck converter.

Stability analysis of discrete-time general homogeneous systems

We propose a new approach for tumor immunotherapy which is based on a switching control strategy defined on domains of attraction of equilibria of interest. For this, we consider a recently derived model which captures the effects of the tumor cells on the immune system and viceversa, through predator-prey competition terms. Additionally, it incorporates the immune systems mechanism for producing hunting immune cells, which makes the model suitable for immunotherapy strategies analysis and design. For computing domains of attraction for the tumor nonlinear dynamics, and thus, for deriving immunotherapeutic strategies we employ rational Lyapunov functions. Finally, we apply the switching control strategy to destabilize an invasive tumor equilibrium and steer the system trajectories to tumor dormancy.

Construction of continuous and piecewise affine Lyapunov functions via a finite-time converse

We provide a procedure which generates a rational control Lyapunov function and a polynomial stabilizer for nonlinear systems described by analytic functions satisfying some regularity conditions. Furthermore, an improved estimate of the domain of attraction of the closed-loop system can be computed by means of previously introduced rational Lypunov functions. For polynomial systems, we indicate that the existence of a polynomial feedback stabilizer is guaranteed by the existence of a rational control Lyapunov function. We illustrate the proposed procedure for the stabilization of the population co-existence equilibrium of a predator-prey model describing tumor dynamics.