On line and double multiplicative integrals

In the present paper the concepts of line and double integrals are modified to the multiplicative case. Two versions of the fundamental theorem of calculus for line and double integrals are proved in the multiplicative case.Publisher's Versio

Partial complete controllability of deterministic semilinear systems

In this paper the concept of partial complete controllability for deterministic semilinear control systems in separable Hilbert spaces is investigated. Some important systems can be expressed as a first order differential equation only by enlarging the state space. Therefore, the ordinary controllability concepts for them are too strong. This motivates the partial controllability concepts, which are directed to the original state space. Based on generalized contraction mapping theorem, a sufficient condition for the partial complete controllability of a semilinear deterministic control system is obtained in this paper. The result is demonstrated through appropriate examples.Publisher's Versio

On complex multiplicative differentiation

In the present paper we discuss multiplicative differentiation for complexvalued functions. Some drawbacks, arising with this concept in the real case, are explained satisfactorily. Some new difficulties, coming from the complex nature of variables, are discussed and they are outreached. Multiplicative Cauchy–Riemann conditions are established. Properties of complex multiplicative derivatives are studied.The paper is a part of B-type project MEKB-09-05Publisher's Versio

On complex multiplicative integration

In the present paper, we extend the multiplicative integral to complex-valued functions of complex variable. The main difficulty in this way, that is, the multi-valued nature of the complex logarithm is avoided by division of the interval of integration to a finite number of local intervals, in each of which the complex logarithm can be localized in one of its branches. Interestingly, the complex multiplicative integral became a multivalued function. Some basic properties of this integral are considered. In particular, it is proved that this integral and the complex multiplicative derivative are bonded in a kind of fundamental theorem.Publisher's Versio

On fractional integral operator over non-Newtonian calculus

The definition of a non-Newtonian calculus is based on the homeomorphism which customary denoted by y = α(x). In the mean of this function, elementary algebraic operations can be modified and we reach to the world of new calculus that is called a Non-Newtonian calculus. Nowadays, fractional operators role an important topic in mathematics because of their applications in many area of interest. In this paper we use an old technique of Cauchy iterated integrals to define biα-fractional integral operator. The allocated method makes the new class of fractional integral operators which are successfully compatible with the non-Newtonian calculi and supported with several examples. Since the non-Newtonian calculi were introduced, the bigeometric calculus has been considered as a brilliant example of these kind of calculi. The definition of fractional integral operator in this calculus leads to Hadamard type fractional integral operator which answers many questions about the behavior of this operator. Classic property of fractional integral operator, semigroup property is stablished and this operator is studied. Moreover, Jensen’s inequality provide boundness theorem for general biα-fractional integral operator.Publisher's Versio

On Partial Complete Controllability of Semilinear Systems

Many control systems can be written as a first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples

Invariant filtering results for wide band noise driven signal systems

Filtering of wide band noise driven systems accounts the following problem. Given an autocovariance function, there are infinitely many wide band noise processes, which have this autocovariance function. Each of them produces its own best estimate. The problem is a selection of the best one of these best estimates. A similar problem arises in control theory as a selection of optimal one of the optimal controls. In this paper we investigate this problem for a wide class of wide band noises. It is proved that in the case of independent wide band and white noises corrupting, respectively, the signal and observations, the best estimates and the optimal controls in the linear filtering and LQG problems are independent of the respective wide band noises. We present a complete set of formulae for the best estimate and, respectively, for the optimal control in terms of the system parameters and autocovariance function of the wide band noise disturbing the signal system.Publisher's Versio

On partial approximate controllability of semilinear systems

In this paper, a sufficient condition for the partial approximate controllability of semilinear deterministic control systems is proved. Generally, the theorems on controllability are formulated for control systems given as a first-order differential equation, while many systems can be written in this form only by enlarging the dimension of the state space. The ordinary controllability conditions for such systems are too strong because they involve the enlarged state space. Therefore, it becomes useful to define partial controllability concepts, which assume the original state space. The method of proof, given in this paper, differs from the traditional proofs by fixed point theorems. The obtained result is demonstrated on examples