Tetsu Shimomura

Near-optimal control of linear multiparameter singularly perturbed systems

In this note, the linear quadratic optimal control for multipa- rameter singularly perturbed systems is studied. The attention is focused on the design of a new near-optimal controller. The resulting controller achieves approximation of the optimal cost. The proposed algorithm has been numerically tested on a real physical example and pro- duced useful results.

Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponents

Let α, ν, β, p and q be variable exponents. Our aim in this article is to deal with Sobolev embeddings for Riesz potentials of order α with functions f in Morrey spaces L Φ,ν,β(G) with Φ(t) = t p (log(e + t)) q over a bounded open set G ⊂ R n . Here p and q satisfy the log-Hölder and the loglog-Hölder conditions, respectively. Also the case when p attains the value 1 in some parts of the domain is included in our results.

SOBOLEV INEQUALITIES FOR ORLICZ SPACES OF TWO VARIABLE EXPONENTS

Our aim in this paper is to deal with Sobolevs embeddings for Sobolev–Orlicz functions with ∇ u ∈ L p (·) log L q (·) (Ω) for Ω ⊂ n . Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results.

Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces

Our aim in this paper is to give Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces. The main result is oriented to the outrange of the well-known Adams theorem.MSC: 31B15, 46E35, 26A33.

Continuity and differentiability for weighted Sobolev spaces

Our aim in this paper is to discuss continuity and differentiability of functions in weighted Sobolev spaces in the limiting case of Sobolevs imbedding theorem.

Numerical algorithm for solving cross-coupled algebraic Riccati equations related to Nash games of multimodeling systems

In this paper, the numerical design of a Nash equilibrium for infinite horizon multiparameter singularly perturbed systems (MSPS) is analyzed. A new algorithm which is based on the Newton method for solving the generalized cross-coupled multiparameter algebraic Riccati equations (GCMARE), is proposed. It is proven that the proposed algorithm guarantees the quadratic convergence. As a result, it is shown the proposed algorithm succeed in improving the convergence rate dramatically compared with the existing results.

Recursive approach of H˞ optimal filtering for multiparameter singularly perturbed systems

Abstract In this paper, we study the H∞ optimal filtering for multiparameter singularly perturbed system (MSPS). In order to obtain the solution, we must solve the multiparameter algebraic Riccati equations (MARE) with indefinite sign quadratic term. First, the existence of a unique and bounded solution of such MARE is newly proven. The main results in this paper are to propose a new recursive algorithm for solving the MARE and to find sufficient conditions regarding the convergence of our proposed algorithm. Using the recursive algorithm, we show that the solution of the MARE converges to a positive semi-definite stabilizing solution with the rate of convergence of O(||μ||i+1).

Hardy Averaging Operator on Generalized Banach Function Spaces and Duality

Let Af(x) := 1 |B(0,|x|)| ∫ B(0,|x|) f(t) dt be the n-dimensional Hardy averaging operator. It is well known that A is bounded on Lp(Ω) with an open set Ω ⊂ Rn whenever 1 < p ≤ ∞. We improve this result within the framework of generalized Banach function spaces. We in fact find the “source” space SX , which is strictly larger than X, and the “target” space TX , which is strictly smaller than X, under the assumption that the Hardy-Littlewood maximal operator M is bounded from X into X, and prove that A is bounded from SX into TX . We prove optimality results for the action of A and its associate operator A′ on such spaces and present applications of our results to variable Lebesgue spaces Lp(·)(Ω) , as an extension of A. Nekvinda and L. Pick [Math. Nachr. 283 (2010), 262–271; Z. Anal. Anwend. 30 (2011), 435–456] in the case when n = 1 and Ω is a bounded interval.

Sobolev inequalities for riesz potentials of functions in \

Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.

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ABSTRACT We shall deal with the boundedness and weak-type boundedness for the generalized fractional integral operators on generalized Orlicz-Morrey spaces of the second kind over non-doubling metric measure spaces in this paper. We also discuss a necessary condition for the boundedness of the generalized fractional integral operators.