Quasi-Newton Embedded Augmented Lagrangian Algorithm for Discretized Optimal Proportional Control Problems

In developing a robust algorithm for solving a class of optimal  control problems in which the control effort  is proportional to the state of the dynamic system, a typical model was studied which generates a constant feedback gain , an estimate of the Riccati for large values of the final time. Involving the third Simpson’s Rule, a discretized unconstrained non-linear problem via the Augmented Lagrangian Method was obtained. This problem was consequently subjected to the Broydon-Fletcher-Goldberg-Shannon(BFGS) based  on Quasi-Newton algorithm. The positive- definiteness of the estimated quadratic control operator was analyzed to guarantee its invertibility in the BFGS. Numerical examples were considered, tested and the results responded much more favourably to the analytical solution with linear convergence. Keywords: Proportional control, feedback gain, Augmented Lagrangian Method, Discretization, BFGS , Simpson’s Rule and Quasi-Newton Metho

On the Discretized Algorithm for Optimal Proportional Control Problems Constrained by Delay Differential Equation

This paper seeks to develop an algorithm for solving directly an optimal control problem whose solution is close to that of analytical solution. An optimal control problem with delay on the state variable was studied with the assumption that the control effort is proportional to the state of the dynamical system with a constant feedback gain, an estimate of the Riccati for large values of the final time. The performance index and delay constraint were discretized to transform the control problem into a large-scale nonlinear programming (NLP) problem using the augmented lagrangian method. The delay terms were consistently discretized over the entire delay interval to allow for its piecewise continuity at each grid point. The real, symmetric and positive-definite properties of the constructed control operator of the formulated unconstrained NLP were analyzed to guarantee its invertibility in the Broydon-Fletcher-Goldberg-Shanno (BFGS) based on Quasi-Newton algorithm. Numerical example was considered, tested and the results responded much more favourably to the analytical solution with linear convergence. Keywords: Simpson’s discretization method, proportional control constant, augmented Lagrangian, Quasi –Newton algorithm, BFGS update formula, delays on state variable, linear convergence

The Growth Enhancement Potentials of Indigenous Plant Growth Promoting Rhizobacteria on Sweet Pepper (Caspicum annum) through Seed Bacterization

The demand for food is rising as the world's population is rising. The second sustainable development goal's objectives of achieving global food security and agricultural sustainability call for swift and coordinated action. The use of traditional mineral fertilizers has increased dramatically in response to the rise in food demand, but this has its drawbacks. Plant growth-promoting rhizobacteria (PGPR) which is a free-living bacteria actively colonize plant roots and aids it growth. They provide an environmentally friendly way to boost plant output. Due to the  high nutritional content and wide range of medicinal applications of sweet pepper (Capsicum annuum L.), it constitute one of the most significant vegetable crops in the world.. sweet pepper (Capsicum annum ) was evaluated through a pot study under the treatment of six bacterial strains. Bacterization of sweet pepper seeds was done via the immersion of seeds in a suspension of bacteria (108 CFU/ml) modified with 2% carboxymethyl cellulose (CMC), The treated seeds were planted in sterilized planting bags with sterilized sand, Effects of bacterized seeds of Capsicum annuum on the shoot height revealed that the highest shoot height was observed at week 8 by isolate AU4(Burkholderia sp) at (9.66±12.15), followed by AU6 (Burkholderia sp ) at week 6 (6.26±0.05). The highest leaf number of Capsicum annuum after bacterization was observed at week 8(9.00±0.0) and 10(9.00±0.0), while the observed control at week 8(6.00±1.0) week 10(7.00±0.0). Plant growth-promoting bacteria applied via seed bacterization has shown potential for increasing the growth parameters of sweet pepper. This could be a formidable substitute for the use of chemical fertilizers. The use of Burkholderia vietnamiensis AU2011 and Burkholderia sp have shown potential for the increment of the agronomical parameter of  C. annuum