Finite plasticity in $$\varvec{P}^\top \! \varvec{P}$$. Part I: constitutive model

We address a finite-plasticity model based on the symmetric tensor \(\varvec{P}^\top \! \varvec{P}\) instead of the classical plastic strain \(\varvec{P}\). Such a structure arises by assuming that the material behavior is invariant with respect to frame transformations of the intermediate configuration. The resulting variational model is lower dimensional, symmetric and based solely on the reference configuration. We discuss the existence of energetic solutions at the material-point level as well as the convergence of time discretizations. The linearization of the model for small deformations is ascertained via a rigorous evolution-\(\Gamma \)-convergence argument. The constitutive model is combined with the equilibrium system in Part II where we prove the existence of quasistatic evolutions and ascertain the linearization limit (Grandi and Stefanelli in 2016).     

Stability and stabilization of a class of fractional-order nonlinear systems for $$\varvec{0<}\,{\varvec{\alpha }} \,\varvec{< 2}$$

This paper investigates the stability and stabilization problem of fractional-order nonlinear systems for \(0<\alpha <2\). Based on the fractional-order Lyapunov stability theorem, S-procedure and Mittag–Leffler function, the stability conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with \(0<\alpha <2\) are proposed. Finally, typical instances, including the fractional-order nonlinear Chen system and the fractional-order nonlinear Lorenz system, are implemented to demonstrate the feasibility and validity of the proposed method.     

Robust {\varvec{L}}_{\varvec{2}}-gain control for a class of cascade switched nonlinear systems

In this paper, we study the robust finite \(L_2 \) -gain control for a class of cascade switched nonlinear systems with parameter uncertainty. Each subsystem of the switched system under consideration is composed of a zero-input asymptotically stable nonlinear part which is a lower dimension switched system, and of a linearizable part. The uncertainty appears in the control channel of each subsystem. We give sufficient conditions under which the nonlinear feedback controllers are derived to guarantee that the \(L_2 \) -gain of the closed-loop switched system is less than a prespecified value for all admissible uncertainty under arbitrary switching. Moreover, we also develop the \(L_2\) -gain controllers for the switched systems with nonminimum phase case.     

Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

Dissipative and generative fractional electric elements in modeling $${\varvec{RC}}$$ RC and $${\varvec{RL}}$$ RL circuits

Nonlinear Dynamics - Two types of constitutive equations consisting of instantaneous and power type hereditary contributions are proposed in order to model generalized capacitor (inductor). The...     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Observer-based quantized sliding mode $${\varvec{\mathcal {H}}}_{\varvec{\infty }}$$ control of Markov jump systems

An adjustable quantized approach is adopted to treat the \(\mathcal {H}_{\infty }\) sliding mode control of Markov jump systems with general transition probabilities. To solve this problem, an integral sliding mode surface is constructed by an observer with the quantized output measurement and a new bound is developed to bridge the relationship between system output and its quantization. Nonlinearities incurred by controller synthesis and general transition probabilities are handled by separation strategies. With the help of these measurements, linear matrix inequalities-based conditions are established to ensure the stochastic stability of the sliding motion and meet the required \(\mathcal {H}_{\infty }\) performance level. An example of single-link robot arm system is simulated at last to demonstrate the validity.     

Existence and Regularity of the Reflector Surfaces in {\mathbb{R}^{n+1}}

In this paper we study the problem of constructing reflector surfaces from the near field data. The light is transmitted as a collinear beam and the reflected rays illuminate a given domain on the fixed receiver surface. We consider two types of weak solutions and prove their equivalence under some convexity assumptions on the target domain. The regularity of weak solutions is a very delicate problem and the positive answer depends on a number of conditions characterizing the geometric positioning of the reflector and receiver. In fact, we show that there is a domain \({\mathcal{D}}\) in the ambient space such that the weak solution is smooth if and only if its graph lies in \({\mathcal{D}}\) .     

Sliding mode control for uncertain active vehicle suspension systems: an event-triggered $$\varvec{\mathcal {H}}_{\infty }$$ H ∞ control scheme

Nonlinear Dynamics - This paper considers the event-triggered sliding mode control problem of uncertain active vehicle suspension systems. A more comprehensive polytope approach is employed to...     

Switching stabilization and { H}_{\varvec{\infty }} performance of a class of discrete switched LPV system with unstable subsystems

Stability and $H_\infty $ performance are analyzed in this paper for a class of discrete switched linear parameter-varying (LPV) systems in which all subsystems’ state-space matrices are parametrically affine, and any subsystem is not stable for parameters varying in a convex set. A switching law is designed to stabilize and satisfy the $H_\infty $ performance of the switched LPV system. By means of the multiple Lyapunov functions method, linear matrix inequality (LMI) conditions for the existence of parameter-dependent Lyapunov functions are proposed. An example shows the effectiveness of the proposed methods.