Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems

According to Newton's second law of motion, we humans describe a dynamical system with a differential equation, which is naturally discretized into a difference equation whenever a computer is used. The differential equation is the physical model in human brains and the difference equation the cyber model in computers for the dynamical system. The physical model refers to the dynamical system itself (particularly, a human-designed system) in the physical world and the cyber model symbolises it in the cyber counterpart. This paper formulates a hybrid model with impulsive differential equations for the dynamical system, which integrates its physical model in real world/human brains and its cyber counterpart in computers. The presented results establish a theoretic foundation for the scientific study of control and communication in the animal/human and the machine (Norbert Wiener) in the era of rise of the machines as well as a systems science for cyber-physical systems (CPS)

On input-to-state stability of stochastic retarded systems with Markovian switching

This note develops a Razumikhin-type theorem on pth moment input-to-state stability of hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), which is an improvement of an existing result. An application to hybrid stochastic delay systems verifies the effectiveness of the improved result

SMC design for robust H∞ control of uncertain stochastic delay systems

Recently, sliding mode control method has been extended to accommodate stochastic systems. However, the existing results employ an assumption that may be too restrictive for many stochastic systems. This paper aims to remove this assumption and present in terms of LMIs a sliding mode control design method for stochastic systems with state delay. In some cases, the proposed method provides a control scheme for finite-time stabilization of stochastic delay systems

On almost sure stability of hybrid stochastic systems with mode-dependent interval delays

This note develops a criterion for almost sure stability of hybrid stochastic systems with mode-dependent interval time delays, which improves an existing result by exploiting the relation between the bounds of the time delays and the generator of the continuous-time Markov chain. The improved result shows that the presence of Markovian switching is quite involved in the stability analysis of delay systems. Numerical examples are given to verify the effectiveness

Stability of hybrid stochastic retarded systems

Abstract-In the past few years, hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), including hybrid stochastic delay systems, have been intensively studied. Among the key results, Mao et al. proposed the Razumikhin-type theorem on exponential stability of stochastic functional differential equations with Markovian switching and its application to hybrid stochastic delay interval systems. However, the importance of general asymptotic stability has not been considered. This paper is to study Razumikhin-type theorems on general theorem moment asymptotic stability of hybrid stochastic retarded systems. The proposed theorems apply to complex systems including some cases when the existing results cannot be used

Stabilisation of hybrid stochastic differential equations by delay feedback control

This paper is concerned with the exponential mean-square stabilisation of hybrid stochastic differential equations (also known as stochastic dierential equations with Markovian switching) by delay feedback controls. Although the stabilisation by non-delay feedback controls for such equations has been discussed by several authors, there is so far little on the stabilisation by delay feedback controls and our aim here is mainly to close the gap. To make our theory more understandable as well as to avoid complicated notations, we will restrict our underlying hybrid stochastic dierential equations to a relatively simple form. However our theory can certainly be developed to cope with much more general equations without any diculty

A GAN-based Tunable Image Compression System

The method of importance map has been widely adopted in DNN-based lossy image compression to achieve bit allocation according to the importance of image contents. However, insufficient allocation of bits in non-important regions often leads to severe distortion at low bpp (bits per pixel), which hampers the development of efficient content-weighted image compression systems. This paper rethinks content-based compression by using Generative Adversarial Network (GAN) to reconstruct the non-important regions. Moreover, multiscale pyramid decomposition is applied to both the encoder and the discriminator to achieve global compression of high-resolution images. A tunable compression scheme is also proposed in this paper to compress an image to any specific compression ratio without retraining the model. The experimental results show that our proposed method improves MS-SSIM by more than 10.3% compared to the recently reported GAN-based method to achieve the same low bpp (0.05) on the Kodak dataset

A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations $\bigg\{ \begin{array}{rl} & u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ & v_x = \nabla_y u.\end{array}$ By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an explicit estimate on the sufficient condition of $h$ to get the existence of solitary waves

Resultados de multiplicidade para sistemas do tipo Schrödinger-Poisson

Doutoramento conjunto em Matemática - Matemática e Aplicações (PDMA)In this thesis, we study the existence and multiplicity of solutions of the following class of Schr odinger-Poisson systems: u + u + l(x) u = (x; u) in R3; = l(x)u2 in R3; where l 2 L2(R3) or l 2 L1(R3). And we consider that the nonlinearity satis es the following three kinds of cases: (i) a subcritical exponent with (x; u) = k(x)jujp2u + h(x)u (4 p < 2 ) under an inde nite case; (ii) a general inde nite nonlinearity with (x; u) = k(x)g(u) + h(x)u; (iii) a critical growth exponent with (x; u) = k(x)juj2 2u + h(x)jujq2u (2 q < 2 ). It is worth mentioning that the thesis contains three main innovations except overcoming several di culties, which are generated by the systems themselves. First, as an unknown referee said in his report, we are the rst authors concerning the existence of multiple positive solutions for Schr odinger- Poisson systems with an inde nite nonlinearity. Second, we nd an interesting phenomenon in Chapter 2 and Chapter 3 that we do not need the condition R R3 k(x)ep 1dx < 0 with an inde nite noncoercive case, where e1 is the rst eigenfunction of +id in H1(R3) with weight function h. A similar condition has been shown to be a su cient and necessary condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity for a bounded domain (see e.g. Alama-Tarantello, Calc. Var. PDE 1 (1993), 439{475), or to be a su cient condition to the existence of positive solutions for semilinear elliptic equations with inde nite nonlinearity in RN (see e.g. Costa-Tehrani, Calc. Var. PDE 13 (2001), 159{189). Moreover, the process used in this case can be applied to study other aspects of the Schr odinger-Poisson systems and it gives a way to study the Kirchho system and quasilinear Schr odinger system. Finally, to get sign changing solutions in Chapter 5, we follow the spirit of Hirano-Shioji, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 333, but the procedure is simpler than that they have proposed in their paper.Nesta tese, estudamos a existência e a multiplicidade de soluções da seguinte classe de sistemas denominada de Schr odinger-Poisson: u + u + l(x) u = (x; u) in R3; = l(x)u2 in R3; onde l 2 L2(R3) ou l 2 L1(R3). Consideram-se não-linearidades que satisfazem um dos seguintes casos: (i) potências que envolvem um expoente sub-cr tico, da forma (x; u) = k(x)jujp2u + h(x)u, (4 p < 2 ), sendo k uma função com sinal indefinido e h uma função positiva; (ii) caso geral de uma não-linearidade indefi nida, da forma (x; u) = k(x)g(u) + h(x)u, sendo k uma função com sinal indefinido e h uma função positiva; (iii) potências que envolvem o expoente crí tico, da forma (x; u) = k(x)juj2 2u + h(x)jujq2u (2 q < 2 ). Convém salientar que esta tese tem três principais inovações, as quais ultrapassam dificuldades geradas pela natureza dos problemas estudados. Primeiro, como um relator anónimo referiu, este é o primeiro trabalho em que se trata a existência de várias soluções de sistemas de Schrödinger- Poisson com não-linearidade indefinida. Segundo, neste estudo encontrou-se um fen ómeno interessante, ver Capítulos 2 e 3, nomeadamente, não ser necess ária a condição R3 k(x)ep 1dx < 0 no caso indefinido e não-coercivo, sendo e1 a função associada ao primeiro valor próprio de + id em H1(R3) com peso h. Note-se que foi demonstrado que uma condi cão semelhante e condição necessária e suficiente na existência de solu cões positivas para equações elíticas semilineares com não-linearidades indefinidas em domínios limitados (ver e.g. Alama-Tarantello, Calc. Var. PDE 1 (1993), 439{475), ou ser uma condição suficiente na existência de soluções positivas para equações elíticas semilineares com não-linearidades indefinidas em RN (see e.g. Costa-Tehrani, Calc. Var. PDE 13 (2001), 159{189). Adicionalmente, o método utilizado pode ser utilizado para estudar outros aspetos dos sistemas de Schrodinger-Poisson, permite também estudar sistemas de Kirchho e sistemas de Schrodinger quasilineares. Por m, para obter soluções com mudança de sinal no Cap. 5, segue se a ideia de Hirano-Shioji, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 333, mas o método utilizado é uma versão simplificada do método apresentado no artigo referido