Volterra series and geometric control theory

It is shown here that controlled differential equations which are analytic in the state and linear in the control have solutions which can be expanded in a Volterra series provided there is no finite escape time. The Volterra kernels are computed in terms of the power series expansion of the functions defining the differential equation. We also give necessary and sufficient conditions for a Volterra series to be realizable by a linear-analytic system. These conditions are particularly easy to test if the Volterra series is finite; a complete theory is worked out for this case. In the final section some applications are considered to singular control, multilinear realization theory, etc.     

On a bounded input-bounded output sufficient condition for nonlinear systems

Using some theorems of the theory of nonlinear Volterra integral equations, a sufficient condition is derived for the boundedness of response of a class of nonlinear control systems. As a consequence, an estimate is given for the upper bound of the response of systems subjected to amplitude limited signals.     

Multilinear Volterra equations of the first kind and some problems of control

Elements of the theory of the multilinear Volterra equations of the first kind relying on the notion of corresponding majorant (integral, differential, functional) equations were set forth. The relation between the simplest majorant equations and special problems of dynamic system control was demonstrated.     

Generalized polynomial operators for nonlinear systems analysis

The concept of the so-called generalized polynomial operators is considered and applied especially to systems described by certain types of nonlinear differential equations. A theorem concerning local invertibility of polynomial operators is given. By an example it is shown how this theorem can be used to prove the existence of solutions, to construct those solutions, and to find a region of BIBO stability of the aforementioned systems. The treatment is quite general, being based on functional analysis. In particular, it can be applied to the systems analyzed by using functional series of Volterra type.     

Integral and integro-differential control plants: Optimality conditions

For the control systems whose dynamics obeys a nonlinear regular integral Volterra equation with additional constraints in the form of equalities, the necessary optimality conditions were established on the basis of the abstract Yakubovich-Matveev theory of optimal control and, in particular, the abstract principle of maximum. Consideration was given to two kinds of the nonlinear controllable singular integral equations with unrestricted multipliers under the integral—with the power kernel of the Cauchy kernel type and with the logarithmic kernel. Attention was paid mostly to the nonlinear controlled dynamic systems obeying an integro-differential Volterra equation of the first order. As before, the study relied on the abstract theory of optimal control. The necessary optimality conditions were established by deriving the corresponding conjugate equation, transversality conditions, and principle of maximum.     

Multilinear Volterra Equations of the First Kind

The Lambert function is used to derive unimprovable estimates for the solutions of nonlinear integral inequalities that play a pivotal role in the study of multilinear Volterra equations of the first kind.     

A new symbolic calculus for the response of nonlinear systems

We present a new operational calculus for computing the response of nonlinear systems to various deterministic excitations. The use of a new tool: noncommutative generating power series, allows us to derive, by simple algebraic manipulations, the Volterra functional series of the solution of a large class of nonlinear forced differential equations. The symbolic calculus introduced appears as a natural generalization to the nonlinear domain, of the well known Heaviside operational calculus. Moreover, this method has the advantage of allowing the use of a computer.     

Computable convergence bounds of series expansions for infinite dimensional linear-analytic systems and application

This paper deals with the convergence of series expansions of trajectories for semi-linear infinite dimensional systems, which are analytic in state and affine in input. A special case of such expansions corresponds to Volterra series which are extensively used for the analysis, the simulation and the control of weakly nonlinear finite dimensional systems. The main results of this paper give computable bounds for both the convergence radius and the truncation error of the series. These results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality. They are available for distributed and boundary control systems. As an illustration, these results are applied to an epidemic population dynamic model. In this example, it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues. The relevance of the method is illustrated by simulations.     

Stochastic functional fourier series, Volterra series, and nonlinear systems analysis

A functional Fourier series is developed with emphasis on applications to the nonlinear systems analysis. In analogy to Fourier coefficients, Fourier kernels are introduced and can be determined through a cross correlation between the output and the orthogonal basis function of the stochastic input. This applies for the class of strict-sense stationary white inputs, except for a singularity problem incurred with inputs distributed at quantized levels. The input may be correlated if it is zero-mean Gaussian. The Wiener expansion is treated as an example corresponding to the white Gaussian input and this modifies the Lee-Schetzen algorithm for Wiener kernel estimation conceptually and computationally. The Poisson-distributed white input is dealt with as another example. Possible links between the Fourier and Volterra series expansions are investigated. A mutual relationship between the Wiener and Volterra kernels is presented for a subclass of analytic nonlinear systems. Connections to the Cameron-Martin expansion are examined as well The analysis suggests precautions in the interpretation of Wiener kernel data from white-noise identification experiments.     

The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay

In this paper, we introduce the optimal control problems governed by a new class of impulsive stochastic partial neutral evolution equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, the analytic semigroup theory, fractional powers of closed operators, and suitable fixed point theorems, we prove an existence result of mild solutions for the control systems in the α-norm without the assumptions of compactness. Next, we derive the existence conditions of optimal pairs of these systems. Finally, application to a nonlinear impulsive stochastic parabolic optimal control system is considered.     

Volterra-series-based equivalent nonlinear system method for subharmonic vibration systems

Subharmonics generation in the nonlinear system, directly using the traditional finite Volterra series, cannot generally represent the aimed system. In this paper, a new approach is presented, which is an extension of single finite Volterra series for representation and analysis of the subharmonic vibration system based on equivalent nonlinear system. The equivalent nonlinear system, which is constructed by pre-compensating the subharmonic vibration system with the super-harmonic nonlinear model, yields the input–output relation between the virtual source and the response of the aimed nonlinear system. Orthogonal least square method is employed to identify the truncated order of Volterra series and predominant Volterra kernels of the equivalent nonlinear system. The MGFRFs (modified generalised frequency response functions) of the equivalent nonlinear system is obtained from the data of the virtual source and response, and verified by comparing the response estimated by the MGFRFs with its true value. Therefore, the aimed subharmonic vibration system can be analysed by taking advantage of a truncated Volterra series based on the equivalent nonlinear system. Numerical simulations were carried out, whose results have shown that the proposed method is valid and feasible, and suitable to apply on representation and analysis of subharmonic vibration systems.     

Volterra series for solving weakly non-linear partial differential equations: application to a dissipative Burgers’ equation

A method to solve weakly non-linear partial differential equations with Volterra series is presented in the context of single-input systems. The solution x(z,t) is represented as the output of a z-parameterized Volterra system, where z denotes the space variable, but z could also have a different meaning or be a vector. In place of deriving the kernels from purely algebraic equations as for the standard case of ordinary differential systems, the problem turns into solving linear differential equations. This paper introduces the method on an example: a dissipative Burgers'equation which models the acoustic propagation and accounts for the dominant effects involved in brass musical instruments. The kernels are computed analytically in the Laplace domain. As a new result, writing the Volterra expansion for periodic inputs leads to the analytic resolution of the harmonic balance method which is frequently used in acoustics. Furthermore, the ability of the Volterra system to treat other signals constitutes an improvement for the sound synthesis. It allows the simulation for any regime, including attacks and transients. Numerical simulations are presented and their validity are discussed.     

On Volterra functional operator games on a given set

The paper is devoted to finding sufficient conditions for an ?-equilibrium in the sense of piecewise-program strategies in antagonistic games associated with nonlinear controlled functional operator equations and cost functionals of a rather general form. The concept of piecewise-program strategies is defined based on the concept of Volterra set chain for operators involved in equations controlled by opponent players. Reduction of distributed control systems to equations of this type is illustrated by examples.     

Perfect bases for differential equations

We present a method for the construction of solutions of certain systems of partial differential equations with polynomial and power series coefficients. For this purpose we introduce the concept of perfect differential operators. Within this framework we formulate division theorems for polynomials and power series. They in turn yield existence theorems for solutions of systems of linear partial differential equations and algorithms to explicitly construct solutions.     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

测度性奇异脉冲微分系统的某些基本理论的研究

具有脉冲解的奇异测度微分系统的基本理论至今尚未建立，本文对这一问题进行了研究，首先建立了奇异测度微分系统与相应的奇异测度积分系统的等性，然后借助地Ｄｒａｚｉｎ逆，在初始条件相容的情况下给出了线性奇异测度微分系统解的表达式，最后得到了一类非性奇异测度微分系统的常数变易公式。     

Collocation method for solving systems of Fredholm and Volterra integral equations

In this paper, a numerical method based on based quintic B-spline has been developed to solve systems of the linear and nonlinear Fredholm and Volterra integral equations. The solutions are collocated by quintic B-splines and then the integral equations are approximated by the four-points Gauss-Turán quadrature formula with respect to the weight function Legendre. The quintic spline leads to optimal approximation and O(h⁶) global error estimates obtained for numerical solution. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods which show that our method is accurate.     

A class of systems of bilinear integral Volterra equations of the first kind of the second order

A class of bilinear systems of integral Volterra equations of the first kind related to the problem of automatic control of a nonlinear dynamic system (object) with unknown structure and vector input and output is studied. Algorithms for an analytic solution to corresponding bilinear systems and its numerical approximation are developed. A special character of the algorithms is illustrated by model examples.     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.     

Existence and Controllability Results for a New Class of Impulsive Stochastic Partial Integro‐Differential Inclusions with State‐Dependent Delay

In this paper, we study a new class of impulsive stochastic partial integro‐differential inclusions with state‐dependent delay in separable Hilbert spaces. Firstly, by using stochastic analysis theory, analytic resolvent operators, fractional powers of closed operators and suitable fixed point theorems, we prove the existence of mild and extremal mild solutions for these systems in the α‐norm. Secondly, we establish the controllability of the controlled stochastic partial integro‐differential inclusions with not instantaneous impulses. The results are obtained under the mixed Lipschitz and Carathéodory conditions. Finally, an example is provided to show the application of our results.