ItemInventory Control with Fixed Cost and Price Optimization in Continuous Time(Wilmington Scientific Publisher) Bensoussan, Alain; Skaaning, S.; Turi, Jànos; Bensoussan, Alain; Turi, JànosWe continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper , we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see ), i.e., the (s; S; p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s; S; p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Selecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy. ItemParameter Dependent Stability/Instability in a Human Respiratory Control System Model(American Institute of Mathematical Sciences, 2013-11) Pradhan, Saroj P.; Turi, Jànos; 0000 0000 4128 774X (Turi, J); 88656618 (Turi, J); Turi, JànosIn this paper a computational procedure is presented to study the development of stable/unstable patterns in a system of three nonlinear, parameter dependent delay differential equations with two transport delays representing a simplified model of human respiration. It is demonstrated using simulations how sequences of changes in internal and external parameter values can lead to complex dynamic behavior due to forced transitions between stable/unstable equilibrium positions determined by particular parameter combinations. Since changes in the transport delays only influence the stability/instability of an equilibrium position a stability chart is constructed in that case by finding the roots of the characteristic equation of the corresponding linear variational system. Illustrative examples are included. ItemStabilization in a State-Dependent Model of Turning Processes(SIAM, 2012-01-03) Hu, Qingwen; Krawcewicz, Wieslaw; Turi, Jànos; 0000 0001 1616 0605 (Krawcewicz, W); 0000 0000 4128 774X (Turi, J); 89645792 (Krawcewicz, W); 88656618 (Turi, J); Hu, Qingwen; Krawcewicz, Wieslaw; Turi, JànosWe consider a two-degree-of-freedom model for turning processes which involves a system of differential equations with state-dependent delay. Depending on process parameters (e.g., spindle speed, depth of cut) the cutting tool can exhibit unwanted vibrations, resulting in a nonsmooth surface of the workpiece. In this paper we propose a feedback law to stabilize the turning process for a large range of system parameters. The feedback law introduces a generic nonhyperbolic stationary point into the model, which generates the main technical challenge of this work. We establish the stability equivalence between the differential equations with state-dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value. Then we show the stability of that nonlinear system with constant delay by computing its normal form. Finally, we obtain conditions on system parameters which guarantee the stability of the state-dependent delay model at the nonhyperbolic stationary point. ©2012 Society for Industrial and Applied Mathematics. ItemGlobal Stability Lobes of Turning Processes with State-Dependent Delay(SIAM, 2012-09-13) Hu, Qingwen.; Krawcewicz, Wieslaw; Turi, Jànos; 0000 0001 1616 0605 (Krawcewicz, W); 0000 0000 4128 774X (Turi, J); 89645792 (Krawcewicz, W); 88656618 (Turi, J); Hu, Qingwen.; Krawcewicz, Wieslaw; Turi, JànosWe obtain global stability lobes of two models of turning processes with inherit nonsmoothness due to the presence of state-dependent delays. In the process, we transform the models with state-dependent delays into systems of differential equations with both discrete and distributed delays and develop a procedure to determine analytically the global stability regions with respect to parameters. We find that the spindle speed control strategy that we investigated in [SIAM J. Appl. Math., 72 (2012), pp. 1–24] can provide essential improvement on the stability of turning processes with state-dependent delay, and furthermore we show the existence of a proper subset of the stability region which is independent of system damping. Numerical simulations are presented to illustrate the general results..