Talha Manzoor

Optimal control for sustainable consumption of natural resources

Abstract In this paper we study optimal policies for a central planner interested in maximizing utility in an economy driven by a renewable resource. It is shown that the optimal consumption path is sustainable only when the intrinsic growth rate of the resource is greater than the social discount rate. The model is formulated as an infinite horizon optimal control problem. We deal with the mathematical details of the problem, develop a precise notion for optimality and establish the existence of optimal control at least when the condition for sustainability is met. We apply the appropriate version of the Pontryagin maximum principle and show a numerical simulation of the optimal feedback law. In the end we present the results along with physical interpretations.

Learning Through Fictitious Play in a Game-Theoretic Model of Natural Resource Consumption

Understanding the emergence of sustainable behavior in dynamic models of resource consumption is essential for control of coupled human and natural systems. In this letter, we analyze a mathematical model of resource exploitation recently reported by the authors. The model incorporates the cognitive decision-making process of consumers and has previously been studied in a game-theoretic context as a static two-player game. In this letter, we extend the analysis by allowing the agents to adapt their psychological characteristics according to simple best-response learning dynamics. We show that, under the selected learning scheme, the Nash Equilibrium is reachable provided certain conditions on the psychological attributes of the consumers are fulfilled. Moreover, the equilibrium solution obtained is found to be sustainable in the sense that no players exhibit free-riding behavior, a phenomenon which occurs in the original open-loop system. In the process, via a Lyapunov-function based approach, we also provide a proof for the asymptotic global stability of the original system which was previously known to be only locally stable.

Optimal Exploitation of Renewable Resources: Lessons in Sustainability from an Optimal Growth Model of Natural Resource Consumption

We study an optimal growth model for a single resource based economy. The resource is governed by the standard model of logistic growth, and is related to the output of the economy through a Cobb-Douglas type production function with exogenously driven knowledge stock. The model is formulated as an infinite-horizon optimal control problem with unbounded set of control constraints and non-concave Hamiltonian. We transform the original problem to an equivalent one with simplified dynamics and prove the existence of an optimal admissible control. Then we characterize the optimal paths for all possible parameter values and initial states by applying the appropriate version of the Pontryagin maximum principle. Our main finding is that only two qualitatively different types of behavior of sustainable optimal paths are possible depending on whether the resource growth rate is higher than the social discount rate or not. An analysis of these behaviors yields general criterions for sustainable and strongly sustainable optimal growth (w.r.t. the corresponding notions of sustainability defined herein).

An extension of the Kalman Filter for a class of measurement models inspired by wide-baseline stereo

We consider the application of the Kalman Filter to systems with a certain type of measurement model. In addition to the state at the current time step, these measurements also depend on the state from one time step earlier. Although such measurement models are not encountered very often, they do appear in some practical control applications in robotic vision. In this paper, we derive a generalized Extended Kalman Filter from a modified form of the basic Bayes filter. The dependence of the measurement model on the previous state is made explicit through the law of total probability, which we include as an additional step in the standard prediction-correction cycle of the Bayes filter. We find that this dependence results in an increase in the measurement noise covariance. We present the mathematical formulation along with the proof of the filter equations. In the end, we demonstrate the filter through a simulation of Monocular SLAM with wide-baseline stereo measurements.

Disparity as a separate measurement in monocular SLAM

In this paper, we investigate the effects of including disparity as an explicit measurement (not just for initialization) in addition to the projective camera measurements, in conventional monocular/bearing-only SLAM in a 2-D world. We conduct an observability analysis for a 1.5-D scenario which theoretically shows that adding disparity measurements influences the observability rank condition, and so disparity can be regarded as a valid measurement. As disparity is dependent on the cameras location one time step earlier, we rederive the EKF equations for measurement models dependent on the previous state. It is shown that this modification results in an increase in the measurement noise covariance. We also discuss the implications of the white noise assumption in EKF being violated with the inclusion of disparity measurements. In the end, we compare the results of a simulated monocular SLAM scenario in a 2-D world with and without disparity measurements. Here we use the magnitude of the eigenvalues of the simulated state covariance matrix as a metric for observability [1]. Our results suggest that including disparity measurements in monocular SLAM improves observability both in the map and pose variables. Hence, this paper is an invitation to consider disparity measurements explicitly, in monocular SLAM.