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The problem of optimal guaranteed cost control for a class of 2D discrete uncertain systems under shift delays and input delays in a Fornasini–Marchesini (FM) second model setting via memory state feedback is considered in this paper. A linear matrix inequality (LMI)-based criterion for the existence of guaranteed cost controllers is established. Furthermore, a convex optimization problem with LMI constraints is formulated to design the optimal guaranteed cost controllers that minimize the upper bound of the closed-loop cost function. Illustrative examples are given to demonstrate the applicability of the proposed method.

Acne vulgaris is a disease of pilosebaceous unit which is characterized by non-inflammatory and inflammatory lesions. Common therapies that are used for the treatment of acne include topical, systemic, hormonal, herbal and combination therapy. So, for the treatment of acne herbal anti acne cream is prepared containing Withania somnifera extract, which is known for its anti-inflammatory action and antioxidant property. Herbal medications are considered safer than allopathic medicines because allopathic medications are associated with different side-effects such as allergy, local irritation, scaling, photosensitivity reaction, itching, peeling, redness etc. The study includes macroscopical and microscopical evaluation of Withania somnifera containing ash value, loss on drying, water extractive value, Fourier Transform Infrared Spectroscopy, Differential Scanning Calorimetry and Ultraviolet Spectroscopy. The prepared formulation was optimized on the basis of in vitro release study. The optimized formulation was evaluated on the basis of greasiness, spreadability, after feel, skin irritancy, viscosity, pH and release kinetics. Anti bacterial study by disc diffusion technique was performed. The result obtained was zone of inhibition of test 25.66 ± 1.12 and 5.89 ± 1.18 at 500 ppm, this proved that optimized formulation has possible anti acne property. So, further investigation such as in vivo studies will help this formulation to be a potential candidate as marketed product.