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This paper is concerned with an inverse problem for a one-dimensional rod through a new method which treats the unknown parameters as time dependent. The method was originally developed for dynamical systems involving ordinary differential equations. With appropriate measurement data the unknown system parameters are guided from an arbitrary initial condition to their true value at a final time. An explicit equation describing the time evolution of the parameter is obtained by minimizing the error along the trajectory. The method leads to an iterative algorithm which is described in detail. A one-dimensional elastic rod whose density is a function of space is considered. Numerical results with the method indicate that accurate estimates of the unknown function can be obtained even in the presence of noise in the data.

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This paper considers adaptive consensus problem on directed graphs of multi-agent systems with unknown control directions. The proposed consensus protocol assumes that each agent receives information about delayed values of state and control signals of respective neighbors. In case of unknown sign of the high-frequency gain of agent dynamics, the proposed algorithm employs Nussbaum-like switching function. The algorithm allows for a designer to select low gain or high gain consensus protocols. Assuming that the underlying graph is strongly connected, it is proved that all agent states converge toward the same value, and the inter-agent coupling parameters are convergent functions. ADAPTIVE CONSENSUS ON DIRECTED GRAPHSThe following theorem examines the stability properties of (27).

This article is concerned with the application of a new method to recover the rate constants in chemical reactions. The method is based on treating the unknown parameters as time dependent. With appropriate experimental data the unknown rate constants are guided from an arbitrary initial condition to their true value at a final time. An explicit equation describing the time evolution of the parameters is obtained by minimizing the error along the trajectory. The method leads to an iterative algorithm which is described in detail. Numerical results with the method indicate that accurate estimates of the rate constants can be obtained directly from experimental data.