Indah Emilia Wijayanti

The Open-Loop Discounted Linear Quadratic Differential Game for Regular Higher Order Index Descriptor Systems

In this paper we consider the discounted linear quadratic differential game for descriptor systems that have an index larger than one. We derive both necessary and sufficient conditions for existence of an open-loop Nash (OLN) equilibrium. In a small macro-economic stabilization game we illustrate that the corresponding optimal response is generically cyclic.

Model predictive control for obstacle avoidance as hybrid systems of small scale helicopter

In this paper, we apply model predictive control for obstacle avoidance of small scale helicopter as a hybrid system such that this UAV can determine the optimal trajectory from initial position to the target position and avoid an obstacle. The hybrid dynamics of this UAV can be interpreted as a piecewise affine (PWA) model that can be transformed into equivalent mixed logical dynamical model (MLD). The PWA model is triggered by state events (coordinate of the UAV for this case). MPC for MLD can be solved using mixed integer quadratic programming (MIQP). We simulate the model and the controller with several shapes of obstacles. From the simulation results, UAV avoids the obstacle optimally using the track generated by MPC.

Distributed model predictive control and application to irrigation canal

Most of the literatures use game approach to design distributed controller such as Feasible Cooperation MPC and Nash-bargaining MPC that contains the weighting for linear convex combination of objective functions. In this paper, we propose a scheme to determine the weighting values that proportional to the control load for each subsystem. To observe the effect of our weighting scheme, we apply the weighting scheme to control the irrigation canal using Feasible-Cooperation MPC and Nash-bargaining MPC. We choose the weighting values are proportional to the volume for each subsystem. We compare the result with weighting that are same for all subsystems. From the numerical simulation results, the total cost of our weighting give better result.

Cooperative dynamic game, discrete time case

In this paper, we study the cooperative dynamic game problem for discrete time case. We solved this problem by determining Pareto solution, continued by finding Nash-bargaining solution. We assume the difference equation in this problem is linear and time invariant. The objective function for each player has the quadratic form and positive definite. We can proof that Pareto solution can be determined by minimizing linear convex combination of all objective functions. The disagreement point of all players is obtained by finding minimax point. The Nash-bargaining solution is selecting a point in Pareto frontier such that the product of utility gains from disagreement point is maximal.

On representation of a ring on a free module over a commutative ring with identity

Let R be a commutative ring with identity and M be a free R-module then we always have a representation of R, that is homomorphism ring μ: R → End R (M), with μ(r) := μ r : M → M and μ r (m) = rm for all r ∈ R and for all m ∈ M. In this paper, we will present some properties of representations of ring R on R-module, based on some notions in representation of R on vector space, such as admissible submodule, equivalence of two representations, decomposable representation and completely reducible representation. It will be shown that if M, N are two free R-modules then two representations μ: R→ End R (M) and : R → End R (N) are equivalent if and only if there is a module isomorphism T : M → N. If R is a principle ideal domain(PID), then it will be shown that every submodule of M is an admissible submodule of M, every representation of ring R on a free R-module is decomposable, and a representation of R on M is completely reducible if and only if M is semisimple.

On representation of polynomial ring on a vector space via a linear transformation

A representation of a ring R is a ring homomorphism from R to the ring of all linear transformations from V to V (End F (V)). From the field F, we can form a polynomial ring F[X]. A representation of F[X] is a ring homomorphism : F[X] → End F (V ) via linear transformation T : V → V with (f(X)) = f(T) for all f(X) ∈ F[X]. For general ring representation ρ: R → End F (V), we have notions of admissibility submodule, and completely reducible and simple admissible submodule. In this paper, we will show that admissible submodules of V are invariant under T, and a representation of F[X] is completely reducible.

*p-MODULES AND A SPECIAL CLASS OF MODULES DETERMINED BY THE ESSENTIAL CLOSURE OF THE CLASS OF ALL *-RINGS

A ring A is called a *-ring if A is a prime ring and A has no nonzero proper prime homomorphic image. The *-ring was introduced by Korolczuk in 1981. Since *-rings have an important role in radical theory of rings, the properties of *-ring have been being investigated intensively. Since every ring can be viewed as a module over itself, the generalization of *-ring into module theory is an interesting investigation. We would like to present the generalization of *-rings in module theory named *p-modules. An A-module M is called a -module if M is a prime A-module and M has no nonzero proper prime submodule. According to the result of our investigation, we show that every *-ring is a *p-module over itself. Furthermore, let A be a ring, let M be an A-module, and let I be an ideal of A with I subset (0:M)A where (0:M)A={a in A| aM={0}}. We show that M is a *p-module over A if and only if M is a *p-module over A/I. On the other hand, the essential closure *k of the class of all *-rings is a special class of rings. As the last result of our investigation, we present the special class of modules determined by *k.

On S(N)-locally prime submodules and the dualization

The primeness of submodules in a module M could be identified by using the fraction of module. On his paper, Jabbar defined an S(N)-locally prime submodule by applying this idea. In this work, we conduct some further properties and a weaker condition of S(N)-locally prime submodules. It is already well known that the dual of prime submodules is second submodules. Hence we also introduce a possibility to dualize the notion of S(N)-locally prime submodules into S(0)-locally second submodules. We investigate the characterizations of locally S(0)-second submodules and give some conditions of rings such that all modules over it will have S(0)-locally second submodules.

Some special classes μ whose upper radical U(μ) determined by μ coincides with the upper radical U(*μ)k determined by (*μ)k

Let μ be a special class of rings, i.e. class of prime associative rings which is closed under ideals and essential extensions. Let *μ be the class of all *μ–rings, that is, rings in μ such that the factor ring is not in μ for every nonzero ideal of R. Apart from their interesting properties, *μ–rings from a generalization of the important class of *–rings introduced and studied in [4], [7], [8], and [14]. France-Jackson and Roux proved useful in describing minimal elements of lattices of hereditary radicals [9].It was observed in [9] that some special classes μ of rings, such as the class of all simple rings with unity, for example, coincide with *μ and the question was asked to describe all such special classes. We will give a partial answer to this question by noting that, in fact, this holds for every special class of simple rings with unity. However, the converse is not true.

Some properties of prime essential rings and their generalization

Prime essential ring and generalized prime essential rings can be used to determine whether a supernilpotent radical is a special radical. Special radical is one of the most important structure in the development of radical theory of rings. Since prime essential rings have an important role in radical theory of rings, we describe some properties of prime essential rings as well as their generalization. We show that every prime essential rings is a subdirectly reducible ring. Furthermore, the prime heart of prime essential ring is zero, so is the semiprime heart of prime an essential ring.