In case the admission to the hospital happens before the policy expiry date and you have duly informed the insurance company before the expiry date and during hospitalization the policy lapses, then the insurance company will honor the claims subject to the terms and conditions of the policy.
If hospitalization happens during the grace period or after policy lapse, as per prevailing rules the insurance company is not expected to admit such claims. However, it is best to inform the company immediately on hospitalization as the decision on claim payment may be discretionary and subject to the circumstances of the case.