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.     

Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\) , and let \(\mathcal {F}\) be a family of subsets of \(S\) . For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\) . The point \(x\) is forward \(\mathcal {F}\) -Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \) . The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\) -nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \) -prolongational limit set \(J\left( x,\mathcal {F}\right) \) . The paper contains applications to control systems.     

Residual Trapping Beneath Impermeable Barriers During Buoyant Migration of \mathrm{CO}_2

The presence of impermeable barriers in a reservoir can significantly impede the buoyant migration of $\mathrm{CO}_2$ injected deep into a heterogeneous geological formation. An important consequence of the presence of these impermeable barriers in terms of the long-term storage of $\mathrm{CO}_2$ is the residual trapping that takes place beneath the barriers, which acts to both increase the storage potential of the reservoir and improve the storage security of the $\mathrm{CO}_2$ . Analytical results for the total amount of $\mathrm{CO}_2$ trapped in a reservoir with an uncorrelated random distribution of impermeable barriers are obtained for both two and three-dimensional cases. In two dimensions, it is shown that the total amount of $\mathrm{CO}_2$ contained in this fashion scales as $n^{5/4}$ , where $n$ is the number of barriers in the vertical direction. In three dimensions, the trapped amount scales as $n^c$ , where $5/4 \le c \le 2$ depending on the aspect ratio of the barriers. The analytical two-dimensional results are compared with results of detailed numerical simulations, and good agreement is observed.     

Stochastic fractional optimal control of quasi-integrable Hamiltonian system with fractional derivative damping

A stochastic fractional optimal control strategy for quasi-integrable Hamiltonian systems with fractional derivative damping is proposed. First, equations of the controlled system are reduced to a set of partially averaged It $\hat{o}$ stochastic differential equations for the energy processes by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and a stochastic fractional optimal control problem (FOCP) of the partially averaged system for quasi-integrable Hamiltonian system with fractional derivative damping is formulated. Then the dynamical programming equation for the ergodic control of the partially averaged system is established by using the stochastic dynamical programming principle and solved to yield the fractional optimal control law. Finally, an example is given to illustrate the application and effectiveness of the proposed control design procedure.     

On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance

A bistable dynamical system with the Duffing potential, fractional damping, and random excitation has been modelled. To excite the system, we used a stochastic force defined by Wiener random process of Gaussian distribution. As expected, stochastic resonance appeared for sufficiently high noise intensity. We estimated the critical value of the noise level as a function of derivative order \(q\) . For smaller order \(q\) , damping enhancement was reported.     

Invariant Parallels, Invariant Meridians and Limit Cycles of Polynomial Vector Fields on Some 2-Dimensional Algebraic Tori in \mathbb R ^3

We consider the polynomial vector fields of arbitrary degree in $\mathbb R ^3$ R 3 having the 2-dimensional algebraic torus $$\begin{aligned} \mathbb T ^2(l,m,n)=\{(x,y,z)\in \mathbb R ^3 : (x^{2l}+y^{2m}-r^2)^2+z^{2n}-1=0\}, \end{aligned}$$ T 2 ( l , m , n ) = { ( x , y , z ) ∈ R 3 : ( x 2 l + y 2 m - r 2 ) 2 + z 2 n - 1 = 0 } , where $l,m$ l , m , and $n$ n positive integers, and $r\in (1,\infty )$ r ∈ ( 1 , ∞ ) , invariant by their flow. We study the possible configurations of invariant meridians and parallels that these vector fields can exhibit on $\mathbb T ^2(l,m,n)$ T 2 ( l , m , n ) . Furthermore, we analyze when these invariant meridians or parallels are limit cycles.     

Liouville Type Property and Spreading Speeds of KPP Equations in Periodic Media with Localized Spatial Inhomogeneity

The current paper is devoted to the study of semilinear dispersal evolution equations of the form $$\begin{aligned} u_t(t,x)=(\mathcal {A}u)(t,x)+u(t,x)f(t,x,u(t,x)),\quad x\in \mathcal {H}, \end{aligned}$$ where $\mathcal {H}=\mathbb {R}^N$ or $\mathbb {Z}^N,\; \mathcal {A}$ is a random dispersal operator or nonlocal dispersal operator in the case $\mathcal {H}=\mathbb {R}^N$ and is a discrete dispersal operator in the case $\mathcal {H}=\mathbb {Z}^N$ , and $f$ is periodic in $t$ , asymptotically periodic in $x$ (i.e. $f(t,x,u)-f_0(t,x,u)$ converges to $0$ as $\Vert x\Vert \rightarrow \infty $ for some time and space periodic function $f_0(t,x,u)$ ), and is of KPP type in $u$ . It is proved that Liouville type property for such equations holds, that is, time periodic strictly positive solutions are unique. It is also proved that if $u\equiv 0$ is a linearly unstable solution to the time and space periodic limit equation of such an equation, then it has a unique stable time periodic strictly positive solution and has a spatial spreading speed in every direction.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression

A recent field theory of elastic bodies undergoing non-smooth submacroscopic geometrical changes (disarrangements) provides a setting in which, for a given homogeneous macroscopic deformation \(F\) of the body, there are typically a number of different states \(G\) of smooth, submacroscopic deformation (disarrangement phases) available to the body. A tensorial consistency relation and the inequality \(\det G\le \det F\) that guarantees that \(F\) accommodates \(G\) determine the totality of disarrangement phases \(G\) corresponding to \(F\) , and it is natural to seek for a given \(F\) those disarrangement phases that minimize the Helmholtz free energy (stable disarrangement phases). We introduce these concepts in the particular context of continuous bodies comprised of many small elastic bodies (elastic aggregates) and in the context where disarrangements do not contribute to the Helmholtz free energy (purely dissipative disarrangements). In this setting, the Helmholtz free energy response \(G\longmapsto \varPsi (G)\) of the pieces of the aggregate determines the totality of disarrangement phases corresponding to \(F\) , which necessarily includes the phase \(G=F\) (compact phase) in which every piece of the aggregate undergoes the given macroscopic deformation \(F\) . When the response function \(\varPsi \) is isotropic and smooth, and when \(\varPsi \) possesses standard semiconvexity and growth properties, the body also admits phases of the form \(G=\zeta _{\min }R\) (loose phases) with \(R\) an arbitrary rotation, provided that \(\zeta _{\min }R \) satisfies the accommodation inequality \(\zeta _{\min }^{3}\le \det F\) . Loose phases, when available, achieve the global minimum \(\varPsi (\zeta _{\min }R)\) of the free energy and consequently are stable and stress-free. When \( \varPsi (G)\) has the specific form \(\varPsi _{\alpha \beta }(G)=(\alpha /2)(\det G)^{-2}+(\beta /2)tr(GG^{T})\) , with \(\alpha \) , \(\beta \) given elastic constants, we determine all of the disarrangement phases corresponding to \(F\) . These include not only the compact and loose phases, but also disarrangement phases \(G\) in which the stress \(D\varPsi (G)\) is uniaxial or planar. Our main result (“stability implies no-tension”) is the assertion that every stable disarrangement phase for \(\varPsi _{\alpha \beta }\) cannot support tensile tractions, and our treatment of elastic aggregates thus provides a natural setting for the emergence of no-tension materials whose response in compression is non-linear. Existing treatments of no-tension materials assume at the outset that the body cannot support tension and that the response in compression is linear.     

Bounded and Unbounded Motions in Asymmetric Oscillators at Resonance

We consider the boundedness and unboundedness of solutions for the asymmetric oscillator $$\begin{aligned} x''+ax^+-bx^-+g(x)=p(t), \end{aligned}$$ where $x^+=\max \{x,0\},x^-=\max \{-x,0\}, a$ and $b$ are two positive constants, $ p(t)$ is a $2\pi $ -periodic smooth function and $g(x)$ satisfies $\lim _{|x|\rightarrow +\infty }x^{-1}g(x)=0$ . We establish some sharp sufficient conditions concerning the boundedness of all the solutions and the existence of unbounded solutions. It turns out that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well-defined functions $\Phi _p(\theta )$ and $\Lambda (h)$ . Some explicit conditions are given for the boundedness of all the solutions and the existence of unbounded solutions. Unlike many existing results in the literature where the function $g(x)$ is required to be a bounded function with asymptotic limits, here we allow $g(x)$ be unbounded or oscillatory without asymptotic limits.     

A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators

Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength \(K_c\) for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between \(\langle K_c\rangle \) and \(N\) , which is \(\langle K_c\rangle \sim \sqrt{N}\) , not only for small \(N\) , but also for large \(N\) . We also investigate the heavy-tailed distribution of \(K_c\) for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of \(K_c\) for large \(N\) from our algorithm to get the empirical density of \(K_c\) . Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with \(N\) and would provide the theoretical support for engineering applications.     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

Semianalytical Solution for \text{ CO}_{2} Plume Shape and Pressure Evolution During \text{ CO}_{2} Injection in Deep Saline Formations

The injection of supercritical carbon dioxide ( $\text{ CO}_{2})$ in deep saline aquifers leads to the formation of a $\text{ CO}_{2}$ rich phase plume that tends to float over the resident brine. As pressure builds up, $\text{ CO}_{2}$ density will increase because of its high compressibility. Current analytical solutions do not account for $\text{ CO}_{2}$ compressibility and consider a volumetric injection rate that is uniformly distributed along the whole thickness of the aquifer, which is unrealistic. Furthermore, the slope of the $\text{ CO}_{2}$ pressure with respect to the logarithm of distance obtained from these solutions differs from that of numerical solutions. We develop a semianalytical solution for the $\text{ CO}_{2}$ plume geometry and fluid pressure evolution, accounting for $\text{ CO}_{2}$ compressibility and buoyancy effects in the injection well, so $\text{ CO}_{2}$ is not uniformly injected along the aquifer thickness. We formulate the problem in terms of a $\text{ CO}_{2}$ potential that facilitates solution in horizontal layers, with which we discretize the aquifer. Capillary pressure is considered at the interface between the $\text{ CO}_{2}$ rich phase and the aqueous phase. When a prescribed $\text{ CO}_{2}$ mass flow rate is injected, $\text{ CO}_{2}$ advances initially through the top portion of the aquifer. As $\text{ CO}_{2}$ is being injected, the $\text{ CO}_{2}$ plume advances not only laterally, but also vertically downwards. However, the $\text{ CO}_{2}$ plume does not necessarily occupy the whole thickness of the aquifer. We found that even in the cases in which the $\text{ CO}_{2}$ plume reaches the bottom of the aquifer, most of the injected $\text{ CO}_{2}$ enters the aquifer through the layers at the top. Both $\text{ CO}_{2}$ plume position and fluid pressure compare well with numerical simulations. This solution permits quick evaluations of the $\text{ CO}_{2}$ plume position and fluid pressure distribution when injecting supercritical $\text{ CO}_{2}$ in a deep saline aquifer.     

Dynamical Parallelepipeds in Minimal Systems

For a topological dynamical system $(X,T)$ ( X , T ) and $d\in \mathbb N $ d ∈ N , the associated dynamical parallelepiped $\mathbf{Q}^{[d]}$ Q [ d ] was defined by Host–Kra–Maass. For a minimal distal system it was shown by them that the relation $\sim _{d-1}$ ～ d ? 1 defined on $\mathbf{Q}^{[d-1]}$ Q [ d ? 1 ] is an equivalence relation; the closing parallelepiped property holds, and for each $x\in X$ x ∈ X the collection of points in $\mathbf{Q}^{[d]}$ Q [ d ] with first coordinate $x$ x is a minimal subset under the face transformations. We give examples showing that the results do not extend to general minimal systems.     

A Comment on the Flow and Heat Transfer Past a Permeable Stretching/Shrinking Surface in a Porous Medium: Brinkman Model

An analytical solution is presented for the boundary-layer flow and heat transfer over a permeable stretching/shrinking surface embedded in a porous medium using the Brinkman model. The problem is seen to be characterized by the Prandtl number $Pr$ , a mass flux parameter $s$ , with $s>0$ for suction, $s=0$ for an impermeable surface, and $s<0$ for blowing, a viscosity ratio parameter $M$ , the porous medium parameter $\Lambda $ and a wall velocity parameter $\lambda $ . The analytical solution identifies critical values which agree with those previously determined numerically (Bachok et al. Proceedings of the fifth International Conference on Applications of Porous Media, 2013) and shows that these critical values, and the consequent dual solutions, can arise only when there is suction through the wall, $s>0$ .     

Miscible and Immiscible Foam Injection for Mobility Control and EOR in Fractured Oil-Wet Carbonate Rocks

Foam injection is a proven enhanced oil recovery (EOR) technique for heterogeneous reservoirs, but is less studied for EOR in fractured systems. We experimentally investigated tertiary \(\text {CO}_{2}\) injections, and \(\text {N}_{2}\) - and \(\text {CO}_{2}\) -foam injections for enhanced oil recovery in fractured, oil-wet limestone core plugs. Miscible \(\text {CO}_{2}\) and \(\text {CO}_{2}\) -foam was compared with immiscible \(\text {CO}_{2}\) - and \(\text {N}_{2}\) -foam as tertiary recovery techniques, subsequent to waterfloods, in fractured rocks with different wettability preferences. At water-wet conditions waterfloods produced approximately 40 % OOIP, by spontaneous imbibition. Waterflood oil recovery at oil-wet conditions was below 20 % OOIP, due to suppressed imbibition where water predominantly flowed through the fractures, unable to mobilize the oil trapped in the matrix. Tertiary, supercritical \(\text {CO}_{2}\) -mobilized oil trapped in the matrix, particularly at weakly oil-wet conditions, by diffusion. Recovery by diffusion was high due to small core samples, high initial oil saturation and a continuous oil phase at oil-wet conditions. Both immiscible \(\text {CO}_{2}\) - and \(\text {N}_{2}\) -foams and miscible, supercritical \(\text {CO}_{2}\) -foam demonstrated high ultimate oil recoveries, but immiscible foam was less efficient (30 pore volumes injected) compared to miscible foam (2 pore volumes injected) to reach ultimate recovery. This is explained by the capillary threshold pressure preventing the injected \(\text {N}_{2}\) gas from entering the matrix, verified by computed X-ray tomography, and the mobilized oil was displaced by the aqueous surfactant in the foam. At miscible conditions, there exists no capillary entry pressure between the oil-saturated matrix and the injected \(\text {CO}_{2}\) , allowing foam to invade the matrix for efficient oil recovery.     

Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation ${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$ where ${\mathcal{O}}$ is a bounded and open domain in ${\mathbb{R}^N, N \geqq 1}$ and W(t) is a Wiener process of the form ${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$ and ${\beta_{k}, k \in \mathbb{N}}$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. One main result established here is that for all initial conditions in ${L^2(\mathcal{O})}$ , it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus, one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows us to prove the finite time extinction of solutions in dimensions ${1\leqq N \leqq3}$ , which is another main result of this work.     

First-principles study of full a-dislocations in pure magnesium

Full a-dislocations on the (0001) basal plane, \((10\bar 10)\) prismatic plane, and \((10\bar 11)\) and \((10\bar 12)\) pyramidal planes in pure magnesium are investigated by using the Peierls-Nabarro model combined with generalized stacking fault (GSF) energies from first-principles calculations. The results show that the \(\left( {10\bar 11} \right)\left\langle {11\bar 20} \right\rangle\) and \(\left( {10\bar 12} \right)\left\langle {11\bar 20} \right\rangle\) slip modes have nearly the same GSF energy barriers, which are obviously larger than the GSF energy barriers of the \(\left( {0001} \right)\left\langle {11\bar 20} \right\rangle\) and \(\left( {10\bar 10} \right)\left\langle {11\bar 20} \right\rangle\) slip modes. For both edge and screw full dislocations, the maximum dislocation densities, Peierls energies, and stresses of dislocations on the \((10\bar 10)\) , (0001), \((10\bar 11)\) , and \((10\bar 12)\) planes eventually increase. Moreover, the Peierls energies and the stresses of screw full dislocations are always lower than those of edge full dislocations for all slip systems. Dislocations on the \((10\bar 11)\) and \((10\bar 12)\) pyramidal planes possess smaller core energies, while the \((10\bar 10)\) prismatic plane has the largest ones, implying that the formation of full dislocations on the \((10\bar 10)\) plane is more difficult.