Ferroelectric materials, such as PZT, PLZT, PMN and BaTiO 3 , provide unique actuator and sensor capabilities for applications including nanopositioning, high speed valves and fuel injectors, camera focusing and shutter mechanisms, ultrasonic devices for biomedical imaging and treatment, and energy harvesting devices. However, to achieve the full potential of the materials, it is necessary to develop and employ models that quantify the creep, rate-dependent hysteresis, and constitutive nonlinearities that are intrinsic to the materials due to their domain structure. The success of models requires that they be highly efficient to implement since real-time applications can require kHz to MHz rates. The calibration of models for specific materials, devices, and applications, requires efficient and robust parameter estimation algorithms. Finally, control designs can be facilitated by models that admit efficient and robust approximate inversion. The homogenized energy model (HEM) is a multiscale, micromechanical framework that quantifies a range of hysteretic phenomena intrinsic to ferroelectric, ferromagnetic and ferroelastic materials. In this paper, we present highly efficient implementation and parameter estimation algorithms for the ferroelectric model. This includes techniques to construct analytic Jacobians and data-driven algorithms to determine initial parameter estimates to facilitate subsequent optimization. The efficiency of these algorithms facilitates material and device characterization and provides the basis for constructing efficient and robust inverse algorithms for model-based control design. The model implementation, calibration, and validation are illustrated using rate-dependent PZT data and single crystal BaTiO 3 data. i Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

Ferroelectric materials, such as lead zirconate titanate, lanthanum-doped lead zirconate titanate, lead manganese niobate, and BaTiO3, provide unique actuator and sensor capabilities for applications including nanopositioning, high-speed valves and fuel injectors, camera focusing and shutter mechanisms, ultrasonic devices for biomedical imaging and treatment, and energy harvesting devices. However, to achieve the full potential of the materials, it is necessary to develop and employ models that quantify the creep, rate-dependent hysteresis, and constitutive nonlinearities that are intrinsic to the materials due to their domain structure. The success of models requires that they be highly efficient to implement since real-time applications can require kilo hertz to mega hertz rates. The calibration of models for specific materials, devices, and applications requires efficient and robust parameter estimation algorithms. Finally, control designs can be facilitated by models that admit efficient and robust approximate inversion. The homogenized energy model is a multiscale, micromechanical framework that quantifies a range of hysteretic phenomena intrinsic to ferroelectric, ferromagnetic, and ferroelastic materials. In this article, we present highly efficient implementation and parameter estimation algorithms for the ferroelectric model. This includes techniques to construct analytic Jacobians and data-driven algorithms to determine initial parameter estimates to facilitate subsequent optimization. The efficiency of these algorithms facilitates material and device characterization and provides the basis for constructing efficient and robust inverse algorithms for model-based control design. The model implementation, calibration, and validation are illustrated using rate-dependent lead zirconate titanate data and single-crystal BaTiO3 data.

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We investigate a mathematical model for the dynamics between tumor cells, immune-effector cells, and the cytokine interleukin-2 (IL-2). In order to better determine under what circumstances the tumor can be eliminated, we implement optimal control theory. We design the control functional to maximize the effector cells and interleukin-2 concentration and to minimize the tumor cells. Next, we show that an optimal control exists for this problem. After which, we characterize our unique optimal control in terms of the solutions to the optimality system, which is the state system coupled with the adjoint system. Finally, we analyze the optimal control and optimality system using numerical techniques.

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