This paper constructs the penalty cost functional for optimizing the two-dimensional control operator of the energized wave equation. In some multiplier methods such as the Lagrange multipliers and Pontrygean maximum principle, the cost of merging the constraint equation to the integral quadratic objective functional to obtain an unconstraint equation is normally guessed or obtained from the fi...

We consider a stochastic linear–quadratic (LQ) problem with possible indefinite cost weighting matrices for the state and the control. An outstanding open problem is to identify an appropriate Riccati-type equation whose solvability is equivalent to the solvability of this possibly indefinite LQ problem. In this paper we introduce a new type of differential Riccati equation, called the generali...

In this paper a very general matrix quadratic equation is considered. This equation is known in literature as asymmetric Algebraic Riccati Equation (ARE) and arises in the solution of many problems in system and control theory and applied mathematics. For this reason a large amount of work has been produced on this topic. We present a new parametrization of the set of solutions of such equation...

Contemporary algebra is self-contradictory. Indeed, it offers the algorithms of solving algebraic equations using either the real or the complex numbers, which are, however, mutually exclusive. For instance, even the simplest quadratic equation c bx ax y 2    in the case of a negative discriminant ac b 4 2  has no solution on the set of real numbers, but does have a solution on the set of c...

A sampling-based optimization method for quadratic functions is proposed. Our method approximately solves the following n-dimensional quadratic minimization problem in constant time, which is independent of n: z∗ = minv∈Rn〈v, Av〉 + n〈v,diag(d)v〉 + n〈b,v〉, where A ∈ Rn×n is a matrix and d, b ∈ R are vectors. Our theoretical analysis specifies the number of samples k(δ, ) such that the approximat...

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as Implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the Implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lift...

a new simple model is introduced for a liquid and an equation of state is derived based on this model and the statistical mechanical calculations. this equation of state works well for the non-polar and slightly polar liquids. the important conclusion that may be deduced from this equation of state is that if the reduced variables of state are chosen and defined appropriately, then, the princip...