This paper addresses the modeling and the control of an autonomous bicycle robot where the reference point is the center of gravity. The controls are based on the wheel heading’s angular velocity and the steering’s angular velocity. They have been developed to drive the autonomous bicycle robot from a given initial state to a final state, so that the total running cost is minimized. To solve the problem, the following approach was used: after having computed the control system Hamiltonian, Pontryagin’s Minimum Principle was applied to derive the feasible controls and the costate system of ordinary differential equations. The feasible controls, derived as functions of the state and costate variables, were substituted into the combined nonlinear state–costate system of ordinary differential equations and yielded a control-free, state–costate system of ordinary differential equations. Such a system was judiciously vectorized to easily enable the application of any computer program written in Matlab, Octave or Scilab. A Matlab computer program, set as the main program, was developed to call a Runge–Kutta function coded into Matlab to solve the combined control-free, state–costate system of ordinary differential equations coded into a Matlab function. After running the program, the following results were obtained: seven feasible state functions from which the feasible trajectory of the robot is derived, seven feasible costate functions, and two feasible control functions. Computational simulations were developed and provided in order to persuade the readers of the effectiveness and the reliability of the approach.

The interactions between the auctioneer (the seller) and the bidders (the buyers) are called a stochastic differential Stackelberg game or a stochastic differential leader-follower game. In this game, the auctioneer (the leader) moves first by setting rules and conventions that govern the auction. Every bidder (the followers) responds by submitting a bid based on the leader's actions (the rules and conventions). The leader aims at maximising his payoff (utility), while every follower aims at maximising their individual utility. The game between the followers may be modelled as a stochastic competitive bidding game, where every follower is rational and aims at maximising their utility function by bidding accordingly.Orientation: This article is related to Finances and Optimisation. The auctioneer designs every auction mechanism such that utility is maximised and cost is minimised.Research purpose: This article proposes an optimal auction mechanism through which auctioneers can assign fairly and efficiently assets to the highest bidders and maximise utility and/or minimise cost.Motivation for the study: One of the tasks of my PhD was about spectrum auction from which I got a vision to design mathematical models and related computational simulations for any asset underlying an auction.Research approach/design and method: Firstly, a study was conducted to model the way auctioneers could analyse and estimate bidders' (buyers') valuations, and then, accordingly, set the prices of the underlying assets or services. An open ascending-bid auction mechanism was also considered. Finally, a first-price sealed-bid auction mechanism for utility maximisation and cost minimisation is investigated. Main findings:The substantive contribution of this article is in the set of mathematical models and computational simulations designed and proposed for the bidders' valuations and the considered open ascending-bid auction. For the investigated first-price sealed-bid auction mathematical models are developed in terms of a combinatorial optimisation problem. The formula computing the expected utility for the auctioneer was designed. Practical/managerial implications:The research provides rigorous ways for optimal auction design to auctioneers and any financial operators or managers. Contribution/value-add:The contributions are in the set of mathematical models and computational simulations. This article models the optimal auction design strategy mechanism as a combinatorial optimisation problem.

This paper aims at computing optimal control policies to drive a self-financing portfolio of financial assets from a given initial financial state to a final state in a given time horizon such that for the first case, the functional portfolio financial risk is minimized and, for the second case, the functional portfolio profit is maximized. The optimal control policies are the optimal investment allocation processes, the optimal state process is the optimal investor’s wealth process, also called the system response to the input control and is obtained by solving the combined system of differential equations formed by the state and costate system of differential equations derived and extracted from Pontryagin’s Minimum Principle. Computational simulations are provided to show the effectiveness and the reliability of the approach.

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Orientation: This article is related to Financial Risk Management, Investment Management and Portfolio Optimisation.Research purpose: The aim is to compute optimal investment allocations from one period to another.Motivation of the study: Financial market systems are governed by random behaviours expressing the complexity of the economy and the politics. Risk Measure and Management are current and major issues for financial market operators and attract the attention of researchers who develop suitable tools and methods to describe and control risk. In this article, financial risk management is considered for an investor operating in the financial market.Research approach/design and method: This research developed Mathematical Models to describe the problem and Computational Simulations to compute, summarise the results and show their reliabilities.Main findings: The results are the investments allocations stored, some tables and the related computational simulations. By going from period one to another, one can notice from the graphs that the portfolio risk is decreasing and the portfolio profit increasing.Practical/managerial implications: The approach used in this article shows a way of solving rigorously any linearly constrained quadratic optimisation problem and any constrained nonlinear problem. It gives the ability of transforming judiciously certain linearly constrained nonlinear programming problems into sequences of linearly constrained quadratic problems and solving them efficiently.Contributions/value-add: This article developed Mathematical Models and Matlab Computer Optimisation Programs to give Computational Simulations. It wrote Computer Programs for a fifth-order autoregressive model to forecast asset profits.