Question

Consider the case of Timmy, a consumer with preferences over oranges (O) and Sauerkraut (S) that give him a utility function U=ln(O) + S.

A. Given that Timmy has an allowance of I and faces prices of Po and Ps, use the Lagrangian Multiplier method to find his optimal consumption of O and S.

B. Demonstrate whether Sauerkraut is a normal or inferior good for Timmy.

C. Find Timmy’s indirect utility function. In a world where I=20, Po = 1 and Ps=2, how much O and S does Timmy buy, and how much utility does Timmy get?

D. Suppose Timmy’s godparents offer to transfer him to an alternate reality where the only difference is the reversal of prices (There Po = 2 and Ps=1). What would this do to Timmy’s consumption of O and S? Would Timmy wish to move to the alternate reality? Show why or why not?  

E. Using the schema – Find Timmy’s Hicksian demand for Oranges? What envelope theorem result did you use?

If Timmy had to pay to enter the alternate reality, what is the most he would pay?

Answer 1

1. Highlights: • This work lays a foundation for studying constraints in quantum control simulations. • The underlying quantum control landscape in the presence of constraints is explored. • Constrained controls can encounter suboptimal traps in the landscape. • The controls are kinematic stand-ins for dynamic time-dependent controls. • A method is developed to transfer between constrained kinematic and dynamic controls. - Abstract: The control of quantum dynamics with tailored laser fields is finding growing experimental success. In practice, experiments will be subject to constraints on the controls that may prevent full optimization of the objective. A framework is presented for systematically investigating the impact of constraints in quantum optimal control simulations using a two-stage process starting with simple time-independent kinematic controls, which act as stand-ins for the traditional dynamic controls. The objective is a state-to-state transition probability, and constraints are introduced by restricting the kinematic control variables during optimization. As a second stage, the means to map from kinematic to dynamic controls is presented, thus enabling a simplified overall procedure for exploring how limited resources affect the ability to optimize the objective. A demonstration of the impact of imposing several types of kinematic constraints is investigated, thereby offering insight into constrained quantum controls

2. On the formulation and numerical simulation of distributed-order fractional optimal control problems

   Science.gov (United States)

   Zaky, M. A.; Machado, J. A. Tenreiro

   2017-11-01

   In a fractional optimal control problem, the integer order derivative is replaced by a fractional order derivative. The fractional derivative embeds implicitly the time delays in an optimal control process. The order of the fractional derivative can be distributed over the unit interval, to capture delays of distinct sources. The purpose of this paper is twofold. Firstly, we derive the generalized necessary conditions for optimal control problems with dynamics described by ordinary distributed-order fractional differential equations (DFDEs). Secondly, we propose an efficient numerical scheme for solving an unconstrained convex distributed optimal control problem governed by the DFDE. We convert the problem under consideration into an optimal control problem governed by a system of DFDEs, using the pseudo-spectral method and the Jacobi-Gauss-Lobatto (J-G-L) integration formula. Next, we present the numerical solutions for a class of optimal control problems of systems governed by DFDEs. The convergence of the proposed method is graphically analyzed showing that the proposed scheme is a good tool for the simulation of distributed control problems governed by DFDE